MATH
Contents
Instructional Focus ............................................................................................................................. iv Developing a Growth Mindset ...................................................................................................... v Whole-Group Activities to Establish a Growth Mindset Classroom ................... vi
DAY 1 PR1ME: Algebraic Equations ......................................................................... 4 Teacher Support . ........................................................................................... 6 Math magazine: Hunt Like a Chameleon ............................................... 8 Teacher Support . ......................................................................................... 10 Practice: Express Yourself . ........................................................................... 32 Practice: A Capital Idea .................................................................................. 32 DAY 2 PR1ME: Using the Guess-and-Check Method ..................................... 11 Teacher Support . ......................................................................................... 13 Math magazine: The Hunt on a Graph . ................................................. 15 Math magazine: Graphing a Line ............................................................. 16 Math magazine: Exit Slips ............................................................................ 17 Practice: Onion Cooking Contest .............................................................. 32 Practice: A Very Cold Day .............................................................................. 32 WEEK 1 Planning & Pacing ............................................................................................... 2 DAY 1 PR1ME: Using the Balance Method .......................................................... 38 Teacher Support . ......................................................................................... 41 Math magazine: The Science of Softball .............................................. 43 Teacher Support . ......................................................................................... 45 Math magazine: Home Run Review ........................................................ 46 Math magazine: Analyzing Word Problems ........................................ 47 Math magazine: Exit Slips ............................................................................ 48 Practice: Expressions Everywhere ........................................................... 64 Practice: Watch Your Step ............................................................................. 64 DAY 2 PR1ME: Practice 3 .............................................................................................. 49 Teacher Support . ......................................................................................... 51 Math magazine: Moon Math ........................................................................ 52 Teacher Support . ......................................................................................... 54 Practice: A Famous Author ........................................................................... 64 Practice: Home, Sweet Home ..................................................................... 64 WEEK 2 Planning & Pacing ............................................................................................. 36 DAY 1 PR1ME: Word Problems ..... ............................................................................ 70 Teacher Support . ......................................................................................... 71 Math magazine: Instagram: Beyond the Square ............................. 72 Teacher Support . ......................................................................................... 74 Practice: What’s the Inequality? ................................................................ 96 Practice: Exploring Underground ............................................................. 96 DAY 2 PR1ME: Practice 4 .............................................................................................. 75 Teacher Support . ......................................................................................... 77 Math magazine: Aspect Ratios Around Us .......................................... 78 Math magazine: Proportional Measures . ............................................. 79 Math magazine: Exit Slips ............................................................................ 80 Practice: Equation Expert ............................................................................. 96 Practice: Born on the Fourth of July ....................................................... 96 WEEK 3 Planning & Pacing ............................................................................................. 68
DAY 3 PR1ME: Practice 2 .............................................................................................. 18 Teacher Support . ......................................................................................... 20 Math magazine: Mapping a Meltdown .................................................. 21 Teacher Support . ......................................................................................... 25 Practice: Watch Your Step ............................................................................. 33 Practice: Time for Fun ..................................................................................... 33 DAY 4 PR1ME: Practice 2 .............................................................................................. 26 Teacher Support . ......................................................................................... 28 Math magazine: Glacier Percentages ..................................................... 29 Math magazine: From Fractions to Percents ..................................... 30 Math magazine: Exit Slips ............................................................................ 31 Practice: X Marks the Spot! .......................................................................... 33 Practice: Iced Tea, Please .............................................................................. 33 DAY 3 PR1ME: Problem Solving ............................................................................... 55 Teacher Support . ......................................................................................... 56 Math magazine: Angles in Orbit ............................................................... 57 Math magazine: Angle Relationships . ................................................... 58 Math magazine: Exit Slips ............................................................................ 59 Practice: Balancing Act ................................................................................... 65 Practice: A Sticky Situation .......................................................................... 65 DAY 4 PR1ME: Word Problems ..... ............................................................................ 60 Teacher Support . ......................................................................................... 61 Math magazine: What’s for Lunch? ......................................................... 62 Practice: Give Y a Try! ...................................................................................... 65 Practice: A Big Group ...................................................................................... 65 DAY 3 PR1ME: Practice 4 .............................................................................................. 81 Teacher Support . ......................................................................................... 83 Math magazine: The Secret Lives of Parrots ...................................... 84 Teacher Support ............................................................................................ 88 Practice: Uncle Sam ......................................................................................... 97 Practice: It’s a Gusher! .................................................................................... 97 DAY 4 PR1ME: Ratio and Fraction ........................................................................... 89 Teacher Support . ......................................................................................... 91 Math magazine: Who’s Visiting the Clay Walls? ................................ 93 Math magazine: Dynamic Data .................................................................. 94 Math magazine: Exit Slips ............................................................................ 95 Practice: Fractions Beyond Compare . .................................................... 97 Practice: Alive and Well .................................................................................. 97
ii Scholar Zone Summer: Math
DAY 1 PR1ME: Solving Word Problems ...... ...................................................... 140 Teacher Support . ...................................................................................... 142 Math magazine: Bringing Math into the Fold ................................. 143 Teacher Support . ...................................................................................... 145 Practice: 8-A ........ ............................................................................................. 170 Practice: A Case from Space ..................................................................... 171 DAY 2 PR1ME: Solving Word Problems ...... ...................................................... 146 Teacher Support . ...................................................................................... 149 Math magazine: Calculating Crease Patterns ................................. 151 Math magazine: The Triangulation Method .................................... 152 Math magazine: Exit Slips ......................................................................... 153 Practice: 8-B ....... .............................................................................................. 170 Practice: Grappling over Grades ............................................................ 173 WEEK 5 Planning & Pacing .......................................................................................... 138 PR1ME: Finding the Number of Times ................................................ 111 Teacher Support . ...................................................................................... 113 Math magazine: Teeming with Life ...................................................... 114 Teacher Support . ...................................................................................... 116 Practice: 6-B ..... ................................................................................................. 128 Practice: The Bad Art Burglary ................................................................ 131 DAY 1 PR1ME: Practice 1 ........................................................................................... 102 Teacher Support . ...................................................................................... 104 Math magazine: A Towering Tradition ................................................ 105 Teacher Support ......................................................................................... 107 Math magazine: What’s the Expression? ........................................... 108 Math magazine: Writing One-Variable Equations ........................ 109 Math magazine: Exit Slips ......................................................................... 110 Practice: 6-A ....... . ............................................................................................. 128 Practice: A Ticklish Tip Problem ............................................................. 129 DAY 2 WEEK 4 Planning & Pacing .......................................................................................... 100 DAY 1 PR1ME: Problem Solving ............................................................................ 182 Teacher Support . ...................................................................................... 186 Math magazine: Real-Life Heroes ......................................................... 190 Teacher Support . ...................................................................................... 192 Math magazine: Area Adventures ......................................................... 193 Math magazine: Dividing Fractions ..................................................... 194 Math magazine: Exit Slips ......................................................................... 195 Practice: 10-A .......... ........................................................................................ 212 Practice: A View From Above .................................................................... 213 DAY 2 PR1ME: Practice 6 ........................................................................................... 196 Teacher Support . ...................................................................................... 198 Math magazine: The Big Pony Swim .................................................... 199 Teacher Support . ...................................................................................... 201 Practice: 10-B .......... ........................................................................................ 212 Practice: A Brief Reply ................................................................................. 215 WEEK 6 Planning & Pacing .......................................................................................... 180
DAY 3 PR1ME: Ratio and Proportion .................................................................. 154 Teacher Support . ...................................................................................... 156 Math magazine: On the Brink . ................................................................ 158 Teacher Support . ...................................................................................... 162 Practice: 9-A ........ ............................................................................................. 170 Practice: Mascot Mischief ........................................................................... 174 DAY 4 PR1ME: Practice 5 ........................................................................................... 163 Teacher Support . ...................................................................................... 166 Math magazine: Releasing Ferrets ....................................................... 167 Math magazine: Writing Percent Equations .................................... 168 Math magazine: Exit Slips ......................................................................... 169 Practice: 9-B ........ ............................................................................................. 170 Practice: The Sweet Tooth Robberies .................................................. 176 DAY 3 PR1ME: Practice 2 ........................................................................................... 117 Teacher Support . ...................................................................................... 119 Math magazine: Reef Areas ...................................................................... 120 Math magazine: Areas of Shaded Regions ....................................... 121 Math magazine: Exit Slips ......................................................................... 122 Practice: 7-A .......... ........................................................................................... 128 Practice: Celebrity (pause) Seating ...................................................... 133 DAY 4 PR1ME: Solving Word Problems ............................................................. 123 Teacher Support . ...................................................................................... 124 Math magazine: Numbers in the News .............................................. 126 Practice: 7-B ........ ............................................................................................. 128 Practice: In a Pickle ....................................................................................... 135 DAY 3 PR1ME: Practice 6 ........................................................................................... 202 Teacher Support . ...................................................................................... 204 Math magazine: Horsey Histograms ................................................... 205 Math magazine: Graphing Histograms .............................................. 206 Math magazine: Exit Slips ......................................................................... 207 Practice: 11-A .......... ........................................................................................ 212 Practice: Carlotta’s Coins ............................................................................ 216 DAY 4 PR1ME: Practice 6 ........................................................................................... 208 Teacher Support . ...................................................................................... 209 Math magazine: Numbers in the News .............................................. 210 Practice: 11-B ........ . ......................................................................................... 212 Practice: Under Particular Conditions ................................................ 217
ANSWER KEY .................................................................................. 219
Grade 7 I Teacher’s Guide iii
Instructional Focus
The Summer School Planning and Pacing Guide provides for six weeks of instruction, four days per week. Further, each day is segmented into three periods, with instruction that includes whole group, small group, and independent/partner work. Note that the bottom row for each week features Growth Mindset language frames for teachers to select from and use throughout the week. Instructional Time Periods Each daily 90-minute instructional block for math is subdivided into three periods. Students rotate through small group and independent work. Times are approximate. Topics: Ratios and Proportional Reasoning, The Number System, Expressions and Equations, Geometry, Statistics and Probability Whole Group: 30 Minutes PR1ME foundational lessons in Algebra, Fractions, and Ratio and Proportion Teacher-Led Small Group: 30 Minutes, 2 Rotations
Read and Apply Math Skills using articles from Math magazine Independent/Partner: 30 Minutes, 2 Rotations Skills Review Additional practice across topics
iv Scholar Zone Summer: Math
Developing a Growth Mindset
The challenges of working with struggling students who lack fundamental math skills and concepts are both a content issue and a mindset issue. Mindset is a relatively new concept brought to light largely through the research of developmental psychologist Carol Dweck. Motivation can be a major challenge for students attending summer school. Growth Mindset refers to the idea that people’s intelligence and abilities can be developed through dedication and hard work. “This view creates a drive for learning and a resilience that is essential for great accomplishment. Virtually all people have these qualities” (Dweck, 2006). Instruction for Week 1 will focus on activities that strive to create a “risk-free” classroom environment where all students are willing to take on challenges and push themselves. Because teacher interaction and feedback play such critical roles in students’ mindsets, each week there are featured instructional language frames designed to make the learning clear, make it safe to risk mistakes, and communicate a high confidence in all students’ ability to rise to the learning challenges. The goal is to give learners feedback about their progress and their results so they can specifically see their growth. Growth Mindset concepts developed during summer school: • Every student has the capacity to grow and learn challenging mathematics. • Effort is far more important than talent when working to master new concepts. • Mistakes, challenges, and setbacks are essential and useful parts of the learning process—especially in mathematics. • The brain is like a muscle. Using your brain makes it stronger.
Grade 7 I Teacher’s Guide v
Whole-Group Activities to Establish a Growth Mindset Classroom
DAY 1: Interest Inventory Conduct an interest survey to establish a community of learners. Read the following four sentence frames, one at a time. Provide time for students to think about their responses—then challenge them to find another student in class who has the same or a similar response. Outside of school, I like to _________________. One special talent I have or someone in my family has is _________________. My favorite website or television show is _________________. One thing that makes me happy is _________________.
Reflect: Read aloud the following questions. Record those interests that are unique to individual students.
What interests do you share with other classmates? What interests or talents are special to you?
DAY 2: What Is Your Mindset? Ask students to think of a time when learning something new was easy for them—when they were able to understand something in only a few hours or days. Have them share how that felt. Then have them describe a time when learning something new was challenging for them—when it took weeks, months, or even years to understand. Compare how that felt. Explain that this summer, students will experience both kinds of feelings, but the important thing to remember is that learning can be frustrating and confusing. It’s normal to feel this way. Explain that practice, work, determination, time, and strategies you use to challenge yourself all contribute to how smart you can become. Tell students that challenging your brain to think mathematically can make you smarter. You can strengthen your brain with an interesting math problem. Present the following problem to the class. If there are 4 students in a group and each person shakes hands with the other members of the group, how many total handshakes are there? Put students in groups of four to try it out and test their predictions. Explain that this is going to be a special summer school experience for them. Together they are going to learn how to decode mathematics and build number sense—much like students decode computer games. Math is all about putting together and taking apart numbers. It’s a way of reasoning and thinking. Mistakes are expected, for that is how we learn.
vi Scholar Zone Summer: Math
DAY 3: Tell Me All You Can Draw a chart like the one below on the board. Invite students to think of their favorite number and be prepared to explain why it is their favorite. Call on a student to name his or her favorite number and explain why it is a favorite. Ask students to raise their hands if they chose the same number. Fill in the chart for that choice. Repeat for a total of five favorite numbers.
Favorite Numbers
Favorite # # of Students
Sample Reason
Reflect How do we use numbers in daily life? What does our list of favorite numbers show about us? (Possible responses: We like a certain athlete; we have many different interests) Do you think your favorite number will change?
DAY 4: I’ll Tell You Discuss the difference between fact and opinion. Facts about a person are things like age, gender, hair or eye color, home address, and number of siblings. Opinions about people include how friendly they are or how well they play a sport or musical instrument. Read aloud the responses to the Reflect statement below and, using a show of thumbs, have students assess whether the statement is classified correctly as a fact or an opinion. If students disagree with one another, ask them to explain their reasoning. Reflect : Most problems in math have only one correct answer. But if you answer incorrectly sometimes, it does not mean that you are bad at math. Instead, it means that you are still learning math. Remind students that a Growth Mindset involves persistence, learning from mistakes, and keeping a positive attitude.
Grade 7 I Teacher’s Guide vii
2
Planning & Pacing
Grade 7 I Week 2
Day 1
Day 2
PR1ME Using the Balance Method to Solve Algebraic Equations Learn: TG pp. 38–40 Teacher Support: TG pp. 41–42
PR1ME Practice 3 Practice: TG pp. 49–50 Teacher Support: TG p. 51
Whole Class
Math magazine The Science of Softball
Math magazine Moon Math
Teacher-Led Small Group
Read and do: TG pp. 43–44 Teacher Support: TG p. 45 Home Run Review, TG p. 46 Analyzing Word Problems, TG p. 47 Exit Slips, TG p. 48
Read and do: TG pp. 52–53 Teacher Support: TG p. 54
Rotate after 30 minutes
Practice Expressions Everywhere; Watch Your Step, TG p. 64
Practice A Famous Author; Home Sweet Home, TG p. 64
Independent/ Partner Work
Growth Mindset Framing When students succeed easily without effort, say: • It’s great that you have that down. Now we need to find something a bit more challenging so you can grow. • It looks like your skills weren’t really challenged by this assignment. Sorry for wasting your time! • You’re ready for something more difficult. • We need to raise the bar for you now.
36 Scholar Zone Summer: Math
Day 3
Day 4
PR1ME Problem Solving Learn: TG p. 55 Teacher Support: TG p. 56
PR1ME Learn Learn: TG p. 60 Teacher Support: TG p. 61
Whole Class
Math magazine Angles in Orbit, TG p. 57 Angle Relationships, TG p. 58 Exit Slips, TG p. 59
Math magazine What’s for Lunch? Read and do: TG pp. 62–63
Teacher-Led Small Group
Rotate after 30 minutes
Practice Balancing Act; A Sticky Situation, TG p. 65
Practice Give Y a Try!; A Big Group, TG p. 65
Independent/ Partner Work
Grade 7 I Teacher’s Guide 37
PR1ME I Whole Class WEEK 2 I DAY 1
Using the balance method to solve algebraic equations Learn a) Solve x + 6 = 10.
b
1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
6
x
x
10
The scale is balanced.
x + 6
=
10
To find the value of x , remove the same number of cubes from both sides until only x is left on one side.
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1
x
x + 6 – 6
=
10 – 6
The scale stays balanced as the same number of cubes have been removed from both sides.
1 1 1 1
x
x
=
4
x = 4 is a solution of x + 6 = 10.
Check: When x = 4 , x + 6 = 4 + 6 = 10
✓
9
ISBN 978-981-09-0495-1
Student Handbook, page 34
38 Scholar Zone Summer: Math
G7_Math_PE_WK1-3 004-067.indd 26
7/23/18 9:57 AM
b) Solve 3 x + 4 = 10.
1 1 1 1
1 1 1 1 1 1 1 1 1 1
3 4 x
x x x
10
3 x + 4
=
10
The scale is balanced.
First, remove the same number of unit cubes from both sides. The scale stays balanced.
1 1 1 1
1 1 1 1
1 1 1 1 1 1
x x x
1 1 1 1 1 1
x x x
3 x + 4 – 4
=
10 – 4
3 x
=
6
1 1
1 1
1 1
x x x
Then, divide each side by 3 to find the value of x .
3 x ÷ 3
=
6 ÷ 3
1 1
x
x
=
2
The scale remains balanced.
x = 2 is a solution of 3 x + 4 = 10.
Check: When x = 2 , 3 x + 4 = 3 × 2 + 4 = 6 + 4 = 10 ✓
10
ISBN 978-981-09-0495-1
Student Handbook, page 35
Grade 7 I Teacher’s Guide 39
G7_Math_PE_WK1-3 004-067.indd 27
7/23/18 9:57 AM
PR1ME I Whole Class (cont.) WEEK 2 I DAY 1
W
1 2
c) Solve
q – 5 = 10.
I can carry out the same operations on both sides until only q is left on one side.
W
1 2 q – 5 + 5 = 10 + 5
Add 5 to both sides
S In
1 2
q = 15
1 2
q × 2 = 15 × 2
Multiply by 2 on both sides
c q
W
1
1 2
1 2
q × 2 =
× 2 × q
= q 1
W
q = 30
q = 30 is a solution of 1 2
q – 5 = 10.
Check: When q = 30 ,
1 2
1 2
× 30 – 5
q – 5 =
= 15 – 5 = 10
✓
1
Practice s 3 1. Use the balance method to solve these equations. Fill in each with +, –, × or ÷. a) j – 14 = 18 Practice 3
2
j – 14 + 14 = 18 +
j =
3
b)
3 f – 19 = 17
= 17
3 f – 19
3 f =
3 f
=
f =
ISBN 978-981-09-0495-1 Student Handbook, page 36
11
JA
40 Scholar Zone Summer: Math
G7_Math_PE_WK1-3 004-067.indd 28
7/23/18 9:57 AM
− Lead students to subtract 6 from both sides of the equation. − Write ‘ x + 6 – 6 = 10 – 6’ on the board. − Relate the subtraction of 6 from both sides of the equation to the second scale pictured. − Have students simplify the equation and see that they get x = 4. − Write ‘ x = 4’ on the board. − Relate this back to the picture of the last scale. − Point out to students that they can check if x = 4 is a solution of the equation by substituting 4 for x in the expression x + 6. Have a student do this and work out the answer on the board. He/she should get an answer of 10. − Conclude that x = 4 is a solution of the equation x + 6 = 10.
Learn
Using the balance method to solve algebraic equations
Learning Outcome: •
Use the balance method to solve an algebraic equation
(a) Stage: Pictorial Representation Relate the activity to the pictures of the scale and the cubes. − Have students look at the algebraic equation in (a) and the first scale. − Guide students to see that they need to ‘remove’ the same number of connecting cubes from both sides of the scale until only the unknown number remains on one side. − Lead them to see that there are 10 cubes on the right and 6 cubes on the left, and that there is 1 box on the left pan that contains an unknown number of cubes, x . − Point out that they can relate the equation to a part-whole bar model as shown in the thought bubble. Guide them to see that x and 6 are both parts of the whole, 10. To find the unknown x , we have to subtract the known part, 6, from the whole, 10. − Together look at the second scale, and lead students to see that in order to have just the box on the left pan, 6 cubes must be removed. − Remind them that if 6 cubes are removed from the left pan, 6 cubes will have to be removed from the right pan, to keep the scale balanced. − Have students look at the last scale. − Point out that after 6 cubes are removed from each side of the scale, there is now 1 box on the left pan and 4 cubes on the right pan. − Lead students to see that since the box of x cubes on the left balances the 4 cubes on the right, there must be 4 cubes in the box. Stage: Abstract Representation Students will see how the algebraic equation relates to the pictorial representation. This association is important as it helps students to visualize the balancing of the scale when they are solving the algebraic equation using the balance method. This method serves as the foundation to solving complex algebraic equations in the later stages.
(b) Stage: Pictorial Representation Relate the activity to the pictures.
− Have students look at the algebraic equation in (b) on PR1ME p. 27 and recall the steps they used in (a) to solve the equation. − Lead students to recognize that they need to ‘remove’ the same number of connecting cubes from both sides of the scale until only the unknown number remains on one side. − Have students see that there are 10 cubes on the right pan and 4 cubes on the left pan. Guide them to also see that there are 3 boxes on the left pan, each containing an unknown number of cubes, x . − Point out that they can relate the equation to a part-whole bar model in the thought bubble. − Have students look at the second scale and lead them to understand that in order to have 3 boxes remain on the left pan, 4 cubes will have to be removed from it. Remind them that when 4 cubes are removed from the left pan, 4 cubes will also have to be removed from the right pan, to keep the scale balanced. − Have students look at the third scale and see that after 4 cubes are removed from each side, 3 boxes remain on the left pan and 6 cubes on the right pan. − Guide students to see that to find the number of cubes in 1 box, we need to divide the left side by 3. We must also divide the right side by 3 to keep the pan balanced. − Have students look at the last scale and see that after dividing both sides by 3, there is 1 box on the left pan and 2 cubes on the right pan. − Lead them to see that since the box of x cubes on the left balances the 2 cubes on the right pan, there must be 2 cubes in the box.
Write ‘Solve x + 6 = 10.’ on the board.
−
− Relate x + 6 = 10 to the first scale pictured on the page. − Have students see that to solve the equation, we must find the value of the unknown letter x that will make the values on both sides of the equal sign the same. − Guide students to see that this is the same as subtracting numbers from both sides of the equation until only the unknown value x is left on one side.
Grade 7 I Teacher’s Guide 41
PR1ME I Whole Class (cont.) WEEK 2 I DAY 1
(c) Stage: Abstract Representation
Stage: Abstract Representations Having gone through the previous stages, students will see how the algebraic equation relates to the pictorial representations.
Students use the balance method to solve algebraic equations that involve fractions. After going through these examples and the two methods of solving algebraic equations, students will realize that even though both methods give the same answer, it is more efficient to use the balance method to solve algebraic equations. Compared to the guess and check method, the balance method usually involves fewer steps.
Write ‘Solve 3 x + 4 = 10.’ on the board.
−
− Relate 3 x + 4 = 10 to the first scale picture on PR1ME p. 27. − Explain that even though there is a number in front of the letter, the steps used to solve the equation are the same as those used in (a). − Remind students that to solve the equation, we must carry out the same operations on both sides of the equation until only the unknown value x is left on one side of the equation. − Guide students to first subtract 4 from both sides of the equation. − Write ‘3 x + 4 – 4 = 10 – 4’ on the board. − Relate the subtraction of 4 from both sides of the equation back to the second scale picture. − Demonstrate how to simplify the equation before moving on to the next step. − Write ‘3 x = 6’ on the board. − Guide students to see that since there is a ‘3’ in front of the x , we need to divide both sides of the equation by 3 in order to find the value of x . − Write ‘3 x ÷ 3 = 6 ÷ 3’ on the board. − Have students look at the third scale picture to relate the division by 3 on both sides of the equation back to the pictorial representation. − Have students simplify the equation and see that they get x = 2. − Write ‘ x = 2’ on the board. − Have students look at the last scale picture. − Highlight to students that they can check if x = 2 is a solution of the equation by substituting 2 for x in the expression 3 x + 4. − Have a student substitute 2 for x in the expression 3 x + 4 and work out the answer on the board. He/she should get an answer of 10. − Conclude that x = 2 is a solution of the equation 3 x + 4 = 10.
1 2 q – 5 = 10.’ on the board.
Write ‘Solve
−
− Guide students in recalling the steps they used in (a) and (b) to solve the equation. − Explain that even though this equation involves fractions, the steps used to solve the equation are the same as those used in (a) and (b). − Remind students that we must carry out the same operations on both sides of the equation until only the unknown value q is left on one side of the equation. − Lead students to first add 5 to both sides of the equation. − Write ‘ 1 2 q – 5 + 5 = 10 + 5’ on the board. − Point out to them to simplify the equation first before moving on to the next step. − Write ‘ 1 2 q = 15’ on the board. − Then, guide students to see that since the fraction involved is 1 2 , we need to multiply both
sides of the equation by 2 in order to find the value of q . Help struggling students to recall that multiplying
1 2 by 2 gives 1. So, multiplying
1 2
q
by 2 will give q .
1 2 q × 2 = 15 × 2’ on the board.
Write ‘
−
− Have students simplify the equation and see that they get q = 30. − Write ‘ q = 30’ on the board. − Remind students that they can check if q = 30 is a solution of the equation by substituting 30 for q in the expression q – 5. Get a student to substitute 30 for q in the expression 1 2 q – 5 and work out the answer on the board. He/she should get an answer of 10. − Conclude that q = 30 is a solution of the equation 1 2 − 1 2 q – 5 = 10.
42 Scholar Zone Summer: Math
Math magazine I Small Group WEEK 2 I DAY 1
Since the 1960s, softball has been mainly played by women.
B ack in 1937, baseball legend Babe Ruth went to bat against a pitcher named John “Cannonball” Baker. But instead of hurling baseballs at the batter, Baker used a larger softball. After swinging and missing several times, Ruth finally told the catcher to switch places with him. “If you’re catching those, you might as well catch them in front of the plate because I can’t hit them,” he said. Why is it so hard for baseball players to hit a softball pitch? The secret is in how the ball is thrown. While baseball pitchers throw the ball overhand, softball pitchers throw underhand. When pitched underhand, a ball moves differently, making it almost impossible to hit without years of practice. Softball was invented in the 1880s as a form of indoor baseball to be played in winter. But people liked the game so SOFTBALL BASEBALL
much, they started playing it outdoors too. Softball has been a spring and summer sport ever since. At first, it was played by both men and women, but because girls weren’t allowed to play baseball, softball was important for female athletes. A softball is larger than a baseball, so it can’t be thrown as quickly. But the distance between the pitcher and batter is shorter, so the ball doesn’t have to travel as far ( see Softball vs. Baseball, below ). As a result, a 70 mph softball pitch reaches home plate in less than 0.4 seconds. That’s slightly faster than a 100 mph pitch reaches a batter in baseball! Throwing the ball underhand also allows softball pitchers to do things that baseball pitchers can’t, such as making the ball curve up at the end.
This is called a “rise ball.” When a ball is thrown overhand, the ball starts high, moving downward on its path to home plate. But with underhand throwing, the ball starts low and travels upward. Experienced softball batters often have a hard time hitting rise balls. So it isn’t surprising that baseball players— even Hall of Famers like Babe Ruth— would struggle. Expecting them to hit home runs would be like expecting a violinist to play the guitar just because both of the instruments have strings. Today there are many places to see softball played at a high level. It’s also a popular high school and college sport. Next year, softball—along with baseball—will return to the Summer Olympics! —Erica Westly
Ball Circumference
There are some important differences between the two sports. Here’s how they compare.
Distance Between Pitcher and Home Plate
Top Pitching Speed
Innings per Game
Student Handbook, page 38
Grade 7 I Teacher’s Guide 43
Math magazine I Small Group (cont.) WEEK 2 I DAY 1
MIXED SKILLS
Use the information in the article and the “Softball vs. Baseball” chart to answer these mixed-skill questions.
1 A combined 25 million Americans play baseball and softball, according to a recent report by the Sports & Fitness Industry Association. With the U.S. population at roughly 325 million, what percent of Americans play softball or baseball, rounded to the nearest percent?
4 The fastest softball players can run from home plate to first base in about 2.5 seconds. Assuming a player kept up that speed, how many seconds would it take them to run from home around all the bases—first, second, and third—and back home again? 5 The path of a softball after it’s hit by a batter can be described by the function f ( x ) = -0.011 x 2 + 1.23 x + 5.5, where x is the softball’s horizontal position in feet and f ( x ), or y , is its vertical height in feet. Could a player standing 80 feet away catch the ball?
2 The formula for the surface area for a sphere is 4 π r 2 . How much larger is the surface area of a softball than a baseball’s, rounded to the nearest hundredth? Use 3.14 for pi. ( Hint: C = 2 π r )
3 In baseball, the distance between the pitcher and home plate is about 60 feet, and the distance between bases is 90 feet. In softball, the field
SOFTBALL FIELD
is smaller, but the ratio of pitching distance to home plate and the distance
between bases is the same: 2:3. Use information from the chart at left and this ratio to label
SOFTBALL BASEBALL 12 inches 9 inches 40 feet 60 feet 70-80 mph 95-105 mph 7 9
the pitcher’s distance to home plate and the distance between bases on the diagram to the right.
te
Student Handbook, page 39
44 Scholar Zone Summer: Math
This lesson plan contains social-emotional learning ( SEL ) competencies related to emotional support and relationship skills.
MIXED SKILLS
3 GUIDED PRACTICE (10 mins) : Choose a student to help you model working through a word problem as a pair. Read the “Your Turn” question 1. Ask your student partner: What do you think we should do to solve this problem? Why? Students can use the following sentence starters to respond to the student partner’s solving strategies: I disagree because . . ., I agree because . . ., I want to expand on . . ., What’s your reasoning for . . ., and more. Then continue to work on solving question 1 with the student in front of the class. After solving, ask your partner: Is there a way that we can check our work? Did we complete all the required steps? 4 GROUP ACTIVITY (20 mins) : Have students work in pairs to solve the remainder of the “Your Turn” problems. Provide each pair with a sheet of chart paper. Have pairs record their work for each problem on the paper. Remind students that each pair should show their work to support their answers and assess reasonableness. Then display these papers around the room so that students can perform a gallery walk. Students will analyze each other’s work for accuracy, discover other methods and strategies for solving the problems, and validate their own answers. 5 ASSESSMENT (10 mins) : Have students SEL EXTENSION Post quotes around the classroom from both male and female athletes from a variety of sports. Have each student stand near one quote with which they strongly agree or strongly disagree. Then have them explain why they chose that quote. Some sample quotes can be found at brightdrops.com/motivational-quotes-for-athletes or sheknows.com/living/slideshow/9277/inspiring-quotes -from-female-athletes . complete the “Home Run Review” +5 skills sheet independently. You can collect these as a formal assessment piece.
The Science of Softball KEY STANDARDS COMMON CORE: 7.NS.A.3 TEKS: 7.3B, *6.3C, *6.3D
MATHEMATICAL PRACTICE: MP1, MP2, MP4 * Additional standards covered in Skills Sheets .
OBJECTIVE Students will use a variety of math skills to analyze and solve word problems about softball and baseball. LESSON PLAN
SKILLS SHEETS: ➜ Analyzing Word Problems
EXIT SLIPS: A On Level B Advanced
➜ Home Run Review
Grade 7 I Teacher’s Guide 45
Math magazine I Small Group (cont.) WEEK 2 I DAY 1
Where Math Gets Real
Home Run Review In “The Science of Softball,” you answered mixed skill questions to learn about softball and baseball. Use what you learned to answer five more questions.
1. A softball game has 7 innings, compared with 9 innings in a baseball game. If each inning in both games lasts an average of 20 minutes, how much longer would a baseball game be than a softball game?
4. Softball was first included as an Olympic sport in 1996. It was a part of the Olympics until 2008 and returned to the Olympics in 2020. If you picked a Summer Olympics between 1996 and 2020 at random, what is the probability that it will include softball? ( Hint: The Summer Olympics are held once every 4 years. )
2. There are 124 countries in the International Softball Federation. Of those countries, 14.5% are located in Africa. How many countries in the International Softball Federation are in Africa?
5. An adult softball bat has a length of 34 inches and a diameter of 2.25 inches. An adult baseball bat has a length of 33 inches and a diameter of 2.5 inches. Using a cylinder as an approximation, which bat has the greater volume, and by how much? Round your answer to the nearest hundredth. Use 3.14 for pi. ( Hint: V = π r 2 h )
3. The longest softball games ever played had the following number of total innings in overtime: 28, 31, 25, 24, 21, 21, 20, 19, 19, 20. What is the mean, median, and mode of this data set?
The Science of Softball > MIXED SKILLS
Student Handbook, page 40
46 Scholar Zone Summer: Math
Analyzing Word Problems Before solving a word problem, you need to analyze the information to understand what you are solving for. You can use the four-step process below to help you analyze and solve word problems. Where Math Gets Real
YOUR TURN ✎
HOW TO ANALYZE A WORD PROBLEM 1 UNDERSTAND: Identify what you are trying to find. What is it asking me to solve for? 2 PLAN: Organize the important information and identify the operations needed. What important information is in the problem? How will I find my answer? What operation(s) can I use? 3 SOLVE: List your strategy and steps to solve, then follow through with your plan. How many steps does the problem require? What equations will help me find the answer? 4 LOOK BACK: Validate your answer by checking all work. Does my answer make sense?
Analyze each word problem below. Then use the boxes to plan out your
strategies and solve. Round your answers to the nearest whole number.
1. During spring training, a softball team scored the following number of runs per game: 9, 3, 5, 10, 8, 10, 12, 3, 6, 5, 8, 8. What is the average number of runs scored?
UNDERSTAND
PLAN
SOLVE
LOOK BACK
2. To score in baseball and softball, the player has to touch each base and return to home plate. In baseball, the total distance traveled by the player is 360 feet. In softball, it’s 240 feet. If both a baseball and a softball team score 7 runs in a game, how much farther will the baseball team have traveled than the softball team?
UNDERSTAND
PLAN
SOLVE
LOOK BACK
3. Clayton Kershaw is a pitcher for the Los Angeles Dodgers, a Major League Baseball team. In 2017, he earned $33 million. In 2018, he earned $33.8 million. What was the percent change in Kershaw’s salary from 2017 to 2018?
UNDERSTAND
PLAN
SOLVE
LOOK BACK
The Science of Softball > OPERATIONS
Student Handbook, page 41
Grade 7 I Teacher’s Guide 47
Math magazine I Small Group (cont.) WEEK 2 I DAY 1
EXIT SLIP A
The Science of Softball > MIXED SKILLS
Where Math Gets Real
1. The World Series is the championship series for Major League Baseball. It was first held in 1903. It is a best- of-seven tournament, which means the lowest number of games played per World Series is 4 and the greatest is 7. Write an inequality to express the range for the number of games played as part of any World Series, using g as the variable. 2. In softball, the pitcher stands in the pitcher’s circle, which is marked on the field in chalk. The radius of the pitcher’s circle is 8 feet. What is the area of the pitcher’s circle? Use 3.14 for pi. ( Hint: Area = π r 2 )
CHECK YOUR UNDERSTANDING:
❑ Want help
❑ Need practice
❑ Almost there
❑ Got it!
EXIT SLIP B
The Science of Softball > MIXED SKILLS
Where Math Gets Real
1. A player’s batting average is found by dividing their number of base hits (hits that let the player advance to a base) by the number of times they were at bat. In the 1991-1992 season, Stacy Cowen had a 0.530 batting average. She was at bat 302 times. How many base hits did she make that season?
2. The shortstop is the softball fielding position between second and third base. The distance between second and third base on a softball field is 60 feet. If the shortstop fields a ball that is directly 12 feet behind third base, what is the diagonal distance the ball will travel from the shortstop to a player standing at second base? Round your answer to the nearest hundredth. ( Hint: Use a 2 + b 2 = c 2 )
CHECK YOUR UNDERSTANDING:
❑ Want help
❑ Need practice
❑ Almost there
❑ Got it!
Student Handbook, page 42
48 Scholar Zone Summer: Math
Check: When q = 30 ,
1 2
1 2
PR1ME I Whole Class WEEK 2 I DAY 2
× 30 – 5
q – 5 =
= 15 – 5 = 10
✓
c) Solve Practice s 3 1. Use the balance method to solve these equations. Fill in each with +, –, × or ÷. a) j – 14 = 18 1 2 q – 5 = 10. 1 2 q – 5 + 5 = 10 + 5 q = 15 Add 5 to both sides Practice 3
I can carry out the same operations on both sides until only q is left on one side.
1 2
1 2
q × 2 = 15 × 2
Multiply by 2 on both sides
j – 14 + 14 = 18 +
j =
1
1 2
1 2
q × 2 =
× 2 × q
= q 1
b)
3 f – 19 = 17
q = 30 3 f – 19
= 17
q = 30 is a solution of 1 2
q – 5 = 10.
3 f =
3 f
=
Check: When q = 30 ,
1 2
1 2
× 30 – 5
q – 5 =
f =
= 15 – 5 = 10
✓
11
© 2015 Scholastic Education International (S) Pte Ltd ISBN 978-981-09-0495-1
1 4
c)
g + 11 = 16
1 4
= 16
g + 11
Practice s 3 1. Use the balance method to solve these equations. Fill in each with +, –, × or ÷. a) j – 14 = 18 Practice 3 1 4 g = 1 4 g =
g =
j – 14 + 14 = 18 +
2. Use the balance method to solve these equations.
j =
a)
q + 32 = 51
b) r – 26 = 26
b)
3 f – 19 = 17
= 17
3 f – 19
3 f =
3 f
=
f =
11
ISBN 978-981-09-0495-1
Student Handbook, page 44
Grade 7 I Teacher’s Guide 49
G7_Math_PE_WK1-3 004-067.indd 34
7/23/18 9:57 AM
Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page 16 Page 17 Page 18 Page 19 Page 20 Page 21 Page 22 Page 23 Page 24 Page 25 Page 26 Page 27 Page 28 Page 29 Page 30 Page 31 Page 32 Page 33 Page 34 Page 35 Page 36 Page 37 Page 38 Page 39 Page 40 Page 41 Page 42 Page 43 Page 44 Page 45 Page 46 Page 47 Page 48Made with FlippingBook - Online catalogs