CoreConnects Mathematics Sample Pages - Grades 6-8

Teacher Resource Manual

Professional Development for College and Career Readiness Teaching a deep understanding of math content and how to use math in the real world

Grades

6-8

SAMPLE

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CoreConnects: Mathematics

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Teaching Math Skills

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to Achieve Common Core Outcomes

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Levels 6-8

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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

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6.RP.A Understand ratio concepts and use ratio reasoning to solve problems . . . . . . . . . . . . . . . 39

6.NS.A Apply and extend previous understandings of multiplication

and division to divide fractions by fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

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6.NS.B Compute fluently with multi-digit numbers and find common factors

and multiples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.NS.C Apply and extend previous understandings of numbers

to the system of rational numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.EE.A Apply and extend previous understandings of arithmetic to algebraic expressions. . . . . . 57

6.EE.B Reason about and solve one-variable equations and inequalities. . . . . . . . . . . . . . . . . . . 61

6.EE.C Represent and analyze quantitative relationships between

dependent and independent variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.G.A Solve real-world and mathematical problems involving area,

surface area, and volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.SP.A Develop understanding of statistical variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 CoreConnects: Mathematics

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CoreConnects: Mathematics – Teaching Math Skills Levels 6-8 – Table of Contents

6.SP.B Summarize and describe distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.RP.A Analyze proportional relationships and use them

to solve real-world and mathematical problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7.NS.A Apply and extend previous understandings of operations

with fractions to add, subtract, multiply, and divide rational numbers . . . . . . . . . . . . . . . . . . 85

7.EE.A Use properties of operations to generate equivalent expressions. . . . . . . . . . . . . . . . . . . 89

7.EE.B Solve real-life and mathematical problems using numerical

and algebraic expressions and equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.G.A Graph points on the coordinate plane to solve real-world and mathematical problems. . . . . . 97

7.G.B Solve real-life and mathematical problems involving angle measure,

area, surface area, and volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

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7.SP.A Use random sampling to draw inferences about a population. . . . . . . . . . . . . . . . . . . . . . 107 7.SP.B Draw informal comparative inferences about two populations . . . . . . . . . . . . . . . . . . . . . 111

7.SP.C Investigate chance processes and develop, use, and evaluate probability models. . . . . . 115

8.NS.A Know that there are numbers that are not rational,

and approximate them by rational numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

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8.EE.A Work with radicals and integer exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.EE.B Understand the connections between proportional

relationships, lines, and linear equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

8.EE.C Analyze and solve linear equations and pairs of simultaneous linear equations. . . . . . . . 133

8.F.A Define, evaluate, and compare functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 8.F.B Use functions to model relationships between quantities. . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8.G.A Understand congruence and similarity using physical models,

transparencies, or geometry software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8.G.B Understand and apply the Pythagorean Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

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CoreConnects: Mathematics – Teaching Math Skills Levels 6-8 – Table of Contents 8.G.C Solve real-world and mathematical problems involving

the volume of cylinders, cones, and spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

8.SP.A Investigate patterns of association in bivariate data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Performance Tasks 6-8 Performance Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Performance Task Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Performance Task Rubric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Performance Task Tracking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Level 6 (1) The Class Trip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Level 6 (2) Road Trip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Level 6 (3) The Birthday Gift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Level 7 (1) Family Matters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Level 7 (2) Home Sweet Home. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Level 7 (3) Physically Fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

Level 8 (1) Movie Madness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Level 8 (2) The Power of Sleep. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Level 8 (3) The New Kid in Town . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

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CoreConnects: Mathematics – Teaching Math Skills Levels 6-8 – Table of Contents

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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.

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» 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.

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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

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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.

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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

7.SP.C 7.SP.C

Objective: Investigate chance processes and develop, use,

and evaluate probability models Pre-Requisite

Current

Growth

Summarize and describe distributions

Investigate chance processes and develop, use, and evaluate probability models

Investigate patterns of association in bivariate data

(6.SP.B.4, 6.SP.B.5)

(7.SP.C.5, 7.SP.C.6, 7.SP.C.7, 7.SP.C.8)

(8.SP.A.1, 8.SP.A.2, 8.SP.A.3, 8.SP.A.4) Performance Task: The Power of Sleep 7.SP.C

Performance Task: Family Matters

Review & Pre-Assessment

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Use the data set to create a frequency table

Answer:

Frequency (1) (4) (4) (8) (3)

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Tally (l) (llll) (llll) (llllllll) (lll)

7.SP.C

Interval 51 – 60 61 – 70 71 – 80 81 – 90 91 – 100

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Data Set: 90, 85, 80, 90, 100, 70, 65, 80, 85, 90, 55, 70, 65, 95, 100, 90, 85, 85, 75, 75,

Describe the distribution of the data by answering the following questions: (20)

2. Which interval is the most common?

(81 – 90)

3. Which interval is the least common?

(51 – 60)

4. What is the range of scores?

(45)

7.SP.C

1. How many test scores are represented in the frequency table?

Instruction UNDERSTAND PROBABILITY •A s a class discuss probability. Guide students to see that it is the chance that an event or outcome will occur. (PS.3)

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7.SP.C

•G uide students to see that they can find the probability of an event by writing the ratio of the number of favorable outcomes to the number of possible outcomes. Discuss the meaning of favorable outcomes.

Discuss how it differs from the possible outcomes. (PS.3)

7.SP.C

7.SP.C

Teaching Math Skills • Provide students with several examples. Ask them to make observations about the probabilities.

Guide students to see that probabilities range from 0, which is impossible, to 1 because the number of favorable outcomes will always be less than the number of possible outcomes, always resulting in a fraction less than 1. (PS.1, PS.3, PS.8)

7.SP.C

• Provide examples of probabilities that are certain and impossible. Have students determine if each event is certain or impossible. (They are possible if they are between 0 and 1) Have students explain how they know. Challenge students to identify the event with the greatest probably that it will occur (the probability closest to 1) and the weakest probability (the probability closest to 0). (PS.1, PS.3, PS.8) • Demonstrate probability using six red crayons and four blue crayons (or comparable manipulatives).

Discuss the number of crayons on the table. Challenge students to write fractions to describe how the crayons are represented. (Red: 6/10; blue: 4/10) (PS.1, PS.2, PS.6)

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•D iscuss and compare the probability of each event. The probability of choosing a red crayon = 0.6, blue = 0.4. Ask students: If the crayons were placed in a bag, and one crayon were drawn at random, would there be greater chance of choosing a red crayon or a blue crayon? (Red) Have students explain their reasoning. (PS.3, PS.7)

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FINDING PROBABILITY

7.SP.C

istribute one die to each student. Discuss how many different numbers there are on the die. Guide •D students to see that the number 6 represents the total number of outcomes. (PS.3, PS.4) •A sk students to determine the probability for rolling a 4. Discuss that if 4 is the desired number, the probability of rolling the 4 is determined by the number of times 4 is on the die. There is only one way to get a 4. So the probability of rolling a 4 is 1 out of 6, or 1/6 = 0.167. (PS.1, PS.2, PS.8)

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• Discuss the probability of rolling an even number. Ask students what information changes in the formula. Ask students to identify how many ways an even number can be rolled. (3) Challenge them to identify the probability of rolling an even number. (0.5) (PS.1, PS.2, PS.3, PS.6)

7.SP.C

• Discuss other events of probabilities from using the die (e.g., rolling an odd number, prime number, multiple of 2, etc.). Work with students to find the probabilities of each event. (PS.1, PS.3, PS.8) • Have each pair of students toss the die 100 times. Have students divvy up so that one partner tosses the die 50 times while the other records the results in a frequency table and vice versa. (PS.4, PS.5, PS.6) • After students have finished rolling the die discuss experimental probability (the probability for the data collected). (PS.3)

7.SP.C

• Challenge students to use this data to find the experimental probability for each previous identified event. Remind students that the total number of times that the die was rolled, or 100, represents the denominator and the number of times a specific event occurred represents the numerator. Complete the experiment’s probability for each die number. (PS.1, PS.2) • Have students calculate the actual probability for each of the events. Start with the denominator.

Ask students how many possible outcomes there are. (6) Remind them that this will represent the denominator. Have students determine the value of the numerator for each event. (PS.1, PS.2) CoreConnects: Mathematics

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Teaching Math Skills

7.SP.C 7.SP.C

• Guide students to compare the experiment and actual probabilities of each event. Students should see that their experimental probabilities are very close to the actual probabilities. If the results are not close, explain possible sources of the discrepancy. (PS.1, PS.4, PS.8) • Repeat the process to explore other probabilities by drawing items out of a bag, flipping a coin, or using tree diagrams. (PS.1, PS.2, PS.4, PS.5, PS.6, PS.7)

Scaffolding Additional Challenge or Rigor

• Allow struggling learners to work with a partner

• P air students with a struggling learner for peer tutoring

•A llow students to use manipulatives such ascounters or tokens when solving problems

• C hallenge students to think of examples of probability in the classroom or school

• Guide students with questioning

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

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•H ave students explore probabilities with only two possible outcomes

7.SP.C

Additional Support

• C hallenge students to explore the probability of compound events

Evidence of Learning

7.SP.C

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ave students identify and explain a real-life •H application of probability

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• Students will understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring where larger numbers indicate greater likelihood.

• Students will be able to approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency.

• Given the probability of an event, students will be able to predict the approximate relative frequency.

• Students will be able to develop a probability model and use it to find probabilities of events 7.SP.C

• Given a model, students will be able to compare probabilities from observed frequencies, and if the agreement is not good, explain possible sources of the discrepancy.

• Students will be able to find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

7.SP.C

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Teaching Math Skills

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Level 7 – Family Matters Adam, Steve, and John are three brothers. They wonder how likely it is that there are other families with three children that all have the same gender.

Part A Create a visual to model all of the gender possibilities of three-children families. Consider all orders of birth.

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What is the probability that all the children will have the same gender?

Gather data representing the gender of the children in 10-20 three-children families.

Part C

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What is the probability that all three siblings in your sample families will have the same gender?

How does the gender of your sample families compare to the probability that you initially discovered?

Would the probability change in your sample families if you collected more samples? Why or Why not?

• Collect more data. How does this affect the probability of your original sample?

What can Adam, Steve, and John conclude about the genders of three children from the same family?

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CoreConnects: Mathematics – Level 7 – Family Matters

Common Core Standard Assessed Skill 7.SP.5 Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

7.SP.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.

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7.SP.8

a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.

Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.

Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.

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Level 7

Answer Key/Teacher Guide

Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

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CoreConnects: Mathematics – Level 7 – Family Matters Adam, Steve, and John are three brothers. They wonder how likely it is that there are other families with three children that all have the same gender.

Part A Create a visual to model all of the gender possibilities of three-children families. Consider all orders of birth.

Teacher: Students should come up with all possibilities of gender distribution in three-children families: BBB, BBG, BGB, BGG, GGG, GGB, GBG, GBB. Eight possible combinations.

Teacher: Out of the eight possibilities, all genders will be the same twice. So the probability is 2 out of 8 or 1/4 or 25%.

Part B Gather data representing the gender of the children in 10-20 three-children families.

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Teacher: Students may use manipulatives (coin, dice (odd/even)), Internet, or classmates to gather data for 10-20 three-children families.

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What is the probability that all three siblings in your sample families will have the same gender?

Teacher: Students will need to find the probability of the samples they came up with. The probability will be the ratio of the same gendered sibling data sets to the total number of items in the sample.

Part C

How does the gender of your sample families compare to the probability that you initially discovered?

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Teacher: Prompt with questions like - How are they similar? How are they different?

Would the probability change in your sample families if you collected more samples? Why or Why not?

• Collect more data. How does this affect the probability of your original sample?

Teacher: Guide students to collect additional data and run the probability again. Discuss how the additional data affected the outcome of probability.

What can Adam, Steve, and John conclude about the genders of three children from the same family?

Teacher: Guide students in a discussion about conclusions they can draw. Answers may vary.

Encourage all answers and questions.

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Level 7

What is the probability that all the children will have the same gender?

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Level 7

CoreConnects: Mathematics – Level 7 – Family Matters

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