Colorado Springs School District 11 ACHIEVEk12
Mathematics







Grade 7,  October Unit


Overview                                                                              
During the last three weeks of quarter 1, 7th graders will use math as a language to explain patterns. Pay close attention to the Essential Questions. You may be asked to answer the questions at the end of this unit. 

Daily Lessons

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2

3

4

5

6

7

8

9

10

11

12

13

14-15

 

Enduring Understandings - important ideas that students should carry with them years beyond the instruction received this year.

  • Numbers can be represented, ordered, and communicated in many different forms.
  • Algebra is a language.
  • Data can be used to draw conclusions and make predictions.
  • Computation, reasoning, and estimation are vital mathematical tools.
  • There are different ways of estimating.
  • Computational strategies are connected to algebraic thinking.

Essential Questions - most important “big picture” questions students should be able to answer after completing learning activities.

  • What are different representations of equivalent numbers?
  • What methods would you use in solving equations?
  • How do you translate between languages?
  • How would you organize and construct displays of data?
  • What conclusions would you make from a variety of data organizers?
  • How would you determine which measure of central tendency best represents a set of data?
  • How would you analyze data and predict future outcomes?
  • How is the order of operations similar to following the rules of a game?
  • How do you decide what strategies to use to solve a problem?
  • How do you decide if the "correct" answer is an appropriate solution?

Standards: Highest Frequency High Frequency Other Standards and E-Skills

Standard 2: (Algebra and Functions) Students use algebraic methods to explore, model and describe patterns and functions involving numbers, shapes, data, and graphs in problem-solving situations and communicate the reasoning used in solving these problems.
2.5a Solve simple equations
2.5b Translate English expressions and algebraic expressions

Standard 5: (Measurement) Students use a variety of tools and techniques to measure, apply the results in problem-solving situations, and communicate the reasoning involved in solving these problems.
5.3 Choose appropriate scale for number lines, graphs, and maps

CSAP Tested Standards  Highest Frequency High Frequency Other Standards and E-Skills

 Highest Frequency = the timing, intensity and level of accountability is extremely high because mastery of these skills will must be demonstrated in multiple test items on CSAP at this grade level.
High Frequency = the timing, intensity and level of accountability is high because mastery of these skills will be tested at this grade level.
 Other Standards and E-Skills = the timing, intensity, and level of mastery are not urgent. It should be introduced during this time so students can experience the concept and return in future quarters to strive towards mastery.

 

Standard 1: Number Sense - October

Read, write, compare, order & place on a number line common fractions (i.e., ½, 1/3, 1/4/ 1/8, 1/10) decimals and percentages.

Read, write, order, locate, and compare positive fractions, and decimal numbers to hundredths (including on a number line).

Recognize pictorial representations of common fractions, decimals and percents.

Show equivalence of common fractions and percents.

Use estimation strategies to determine reasonable answers in addition and subtraction.

Compare commonly used proper fractions and terminating decimals.

Use number sense to estimate sums and differences of fractions and decimals (i.e., 5/6 + 7/8 must be equal to an amount less than 2, since each fraction is less than 1).

Standard 2: Patterns and Algebra - October

Recognize, explain and extend a linear pattern in a problem solving situation.

Analyze and describe patterns using words, tables and graphs.

Standard 3: Data and Graphs - October

Organize and construct displays of data including tables, charts, pictographs, line plots, bar graphs, circle graphs and line graphs from a given set of data using appropriate tools.

Analyze and draw conclusions about tables, charts, line graphs, bar graphs, pictographs, and line plots.

Read data from tables, charts, pictographs, line plots, bar graphs, circle and line graphs with categorical (i.e., number of pets) and numerical (i.e., heights of buildings) data.

Describe how data collections methods affect the nature of a data set.

Distinguish between median and mode of a set of data.

Identify mean, median, mode and range of data.

Standard 4: Geometry - October

Identify, compare, and analyze two-dimensional shapes by attributes (acute angles, obtuse angles, right angles, parallel lines, perpendicular lines, intersecting lines, congruence, similarity, rays, lines, line segments, etc.)

Standard 5: Measurement - October

Select and use the appropriate unit and tool to measure to the degree of accuracy required in a particular problem (i.e., degree of angle, nearest ¼ inch).

Determine the appropriate unit of measure (metric or US customary) when estimating distance, capacity, weight, and temperature.

Generate a hypothesis or conjecture and solve measurement problems of appropriate complexity created from real-world problems.

Standard 6: Computation - October

Using number sentences and any of the four basic operations, create and illustrate “real” problems with whole numbers.

Use, explain, and demonstrate the understanding of the four basic operations in problem-solving situations.

Use and explain a variety of estimation techniques to solve problems.

Using concrete materials or pictures, determine commonly-used percentages (i.e., 25%, 50%, etc.) in problem-solving situations.

Determine whether information given in a problem is sufficient, insufficient, or extraneous.

Lessons

Lesson 1: Reporting Frequency and Using Line Graphs
 Duration: 1 day
Materials Needed: Prentice Hall Mathematics 7, Course 2 

Activities

  1. There are many ways to organize data so that you can analyze it. The simplest ways are using frequency tables – marking a chart with tally marks then counting them, or line plots – marking a graph with x for each time an item in a set of data occurs. 

  2. You can easily translate the data from either a frequency table or a line plot into a histogram – essentially a bar graph with no spaces between the bars. There are excellent examples of all three data displays on pp. 573 – 575.

  3. Go to the Shodor.org Website and scroll down to the statistics section, and click on the Histogram link. Complete the activity.

  4. You should read Section 11-1, pp 573 - 576 in your text. Use the Instant Check system to be sure you understand the material you read.  To do this, solve the problems beside the Check Understanding marks in each section, then check your answer with the key at the back of the book, beginning on p. 755. If you miss a problem, reread the example to clarify the concept.

  5. After you read the material, practice your work. On pp. 576 - 578, do problems # 2, 14, 21, and 23.
    If you found these problems difficult or you made mistakes, do #      4, 7, 8, 12, 15, and 16 – 20.
    If you got them all correct and found them easy, do #      15 – 20, 24, and 25.

Assessment:  Complete pg. 578 # 26 – 28. 

Differentiation
Extension:  Investigation: Collecting Data on p. 576.  Instead of asking 30 classmates, ask 30 people with whom you are in occasional contact.  Compare the data of the first 10 to the data of the whole sample.  Which is more representative of a larger group?  Why?
Support:  Review Lesson 1-6 and do problems 1 – 4 on p. 573
Do additional problems on pp. 576 - 578 needed.
For extra practice, solve # 1 - 2 in the Extra Practice section on p. 694. 

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Lesson 2: Spreadsheets and Data Displays
 Duration: 1 day
Materials Needed: Prentice Hall Mathematics 7, Course 2 
Access to a spreadsheet program such as Excel for extension activity

Activities

  1. There are many ways to organize data so that you can analyze it.  Another way to organize and analyze data is the spreadsheet.  These days, most spreadsheets are generated on computers.  You can enter data in the cells then, after the data is analyzed, use the graphing tool to create the most appropriate data display.

  2. Double line and double bar graphs compare two sets of data, for instance the results of a survey taken at two different times. You should read Section 11-2, pp 579 - 581 in your text.  Use the Instant Check system to be sure you understand the material you read.  To do this, solve the problems beside the Check Understanding marks in each section, then check your answer with the key at the back of the book, beginning on p. 755.  If you miss a problem, reread the example to clarify the concept.

  3. After you read the material, practice your work.  On pp. 581 - 584, do problems # 5, 16, 24, 31, and 32.
    I    f you found these problems difficult or you made mistakes, do #  2 – 14 even, 17 – 22, and 25 - 28.
    If you got them all correct and found them easy, do #      11 – 22,  25 – 28, and 33 – 34.

Assessment: Complete pg. 584 # 38 – 41.  

Differentiation
 Extensions:  Investigation: Making Graphs on p. 581.  You will need access to a computer spreadsheet program like Excel to complete this assignment.  Instead of finding data about students in a specific school, go to www.d11.org and collect data about students in either different grades or different schools in Colorado Springs School District 11 for your spreadsheet.  What does the data tell you about how the population in the district has changed over the last 5 years?
Support:  Do additional problems on pp. 581 – 584 as needed.
For extra practice, solve # 3 – 5 in the Extra Practice section on p. 694

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Lesson 3: Organizing and Analyzing Data
 Duration:  1 day
Materials Needed:  Prentice Hall Mathematics 7, Course 2

Activities

  1. There are many ways to organize data so that you can analyze it.  Additional ways to organize data are stem-and-leaf plots and box-and-whisker plots.  For either plot, it is easiest to order your data from smallest to largest BEFORE you create the plots.

  2. Stem-and-leaf plots break data values into stems – digits on the left, and leaves – digits on the right.   You can easily compare two sets of data with a back-to-back stem-and-leaf plot whose stems are the same.  There is a good example of this on p. 586.

  3. Box-and-whisker plots are used to display data in a range.  They are often used in large scientific studies so that the scientists can see the range of the data collected.

  4. You should read Section 11-3, pp 585 - 587 in your text.  Use the Instant Check system to be sure you understand the material you read.  To do this, solve the problems beside the Check Understanding marks in each section, then check your answer with the key at the back of the book, beginning on p. 755.  If you miss a problem, reread the example to clarify the concept.

  5. After you read the material, practice your work.  On pp. 587 - 590, do problems # 4, 11, 23, and 25.
    If you found these problems difficult or you made mistakes, do #  2, 5 – 7, 9, 12 – 18, and 24.
    If you got them all correct and found them easy, do #  3, 10, 12 – 14, 19 – 22, 26, and 28.

Assessment: Complete pg. 589 # 28 – 32. 

Differentiation
Extensions:  Write two short paragraphs describing how each type of plot is constructed, what each shows, and how it can be used.  Include specific examples from real world applications.
Support:  Review Lesson 1-10 and do # 1- 6 on p. 585. Do additional problems on pp. 587 – 590 as needed. For extra practice, solve # 6 – 7 in the Extra Practice section on p. 694

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Lesson 4: Using Data to Persuade
 Duration: One class period
Materials Needed:  Prentice Hall Mathematics 7, Course 2

Activities

  1. Graphs and statistics are powerful tools and have persuasive powers.  How that data is presented can affect a reader’s impressions about the meaning of the data.

  2. If the results of a survey of 20,000 people are very close, using a scale that has increments of 500 people can demonstrate that.  If, however, you want to emphasize the differences, you might use a scale of 10, with a break between 10 and 10,000.  That could mislead your audience.  See the graph on p. 611 for an example of this.

  3. You can also mislead by your presentation of mean, median, and mode.  Remember that outliers can skew the mean!
    You should read Section 11-7, pp 606 - 608 in your text.  Use the Instant Check system to be sure you understand the material you read.  To do this, solve the problems beside the Check Understanding marks in each section,     then check your answer with the key at the back of the book, beginning on p. 755.  If you miss a problem, reread the example to clarify the concept.

  4. After you read the material, practice your work.  On pp. 608 - 611, do problems 2 – 18 even. Now, look at periodicals and find 3 examples of misleading graphs or statistics.  Cut them out, and, paste a paragraph explaining why it is misleading on the back of each.

Assessment:  Complete pg. 611 # 23 – 27.  

Differentiation
Extensions:  Read and complete p. 612, “Writing to Persuade”
Support:  Review Lesson 1-10 and do # 1- 4 on p. 606.
Do additional problems on pp. 587 – 590 as needed.
For extra practice, solve # 6 – 7 in the Extra Practice section on p. 694.

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Lesson 5:  Divisibility Tests
 Duration: One class period
Materials Needed: Prentice Hall Mathematics 7, Course 2

Activities

  1. A whole number is divisible by another whole number if the remainder is zero when you divide one by the other.
     10 / 2 = 5
    Since the remainder is 0, 10     is divisible by 2.  Remember that it is also divisible by the quotient – by 5!

  2. There are a number of divisibility tests listed on pp. 141 and 142.  Make a poster with the tests listed.  There are two to add to the list: A whole number is divisible by 6 if it is divisible by 2 and by 3.
    A whole number is divisible by 100 if it ends in 2 zeroes.

  3. Read Section 3-3, pp 141 - 142 in your text.  Use the Instant Check system to be sure you understand the material you read.  To do this, solve the problems beside the Check Understanding marks in each section, then check your answer with the key at the back of the book, beginning on p. 755.  If you miss a problem, reread the example to clarify the concept.

  4. After you read the material, practice your work.  On pp. 143 - 144, do problems # 17, 22, 27, 28, and 30.
    If you found these problems difficult or you made mistakes, do #      1 – 15 odd, 18 – 21, and 35 - 38.
    If you got them all correct and found them easy, do #      31 - 41.

Assessment: Complete pg. 144 # 42 – 45. 

Differentiation
Extensions:  Try to determine divisibility rules for 7, 11, 20, and 50.  Can you find any patterns?  Is there a relationship between divisibility rules and prime numbers?
Support:  Review Lesson 1-8 and do # 1- 6 on p. 141.
Do additional problems on pp. 143 – 144 as needed.
For extra practice, solve # 10 – 14 in the Extra Practice section on p. 686

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Lesson 6:  Prime Factorization
 Duration: One class period
Materials Needed: Prentice Hall Mathematics 7, Course 2 

Activities

  1. Prime numbers are whole numbers greater than 1 which have exactly two factors – 1 and the number itself.  Composite numbers are whole numbers greater than 1 which have more than two factors.
    The factors of 11 are 1 and 11, so 11     is prime.
    The factors of 12 are 1, 2, 3, 4, 6, and 12,      so 12 is composite. 
    You can use your divisibility test rules to help figure out if a number is prime or composite.  Of course, there is always the problem of 7. We use factor trees to find all of the prime factors of a number.

  2. You can compare the factors of two numbers to find the greatest common factor (GCF). Find the GCF of 120 and 12.          
    12 = 2     · 2 · 3  = 22 · 3
    120 = 2    3 · 3 · 5

    The common factors are 2     · 2 · 3, which equals 12, so the GCF is 12.

         120 is divisible by 2
     2 x 60  2 is prime, so circle that.  60 is divisible by 2, so continue.
    2 x 30  2 is prime, so circle that.  30 is divisible by 2, so continue.
    2 x 15  2 is prime, so circle that.  15 is divisible by 5, so continue.
    3 x  5   Both 3 and 5 are prime, so circle them.
         The prime factors of 120 are 2 · 2 · 2 · 3 · 5.  You can use exponents to shorten that to 23 · 3 · 5.
    You can compare the factors of two numbers to find the     greatest common factor (GCF).

  3. Read Section 3-4, pp 145 - 147 in your text.  Use the Instant Check system to be sure you understand the material you read.  To do this, solve the problems beside the Check Understanding marks in each section, then check your answer with the key at the back of the book, beginning on p. 755.  If you miss a problem, reread the example to clarify the concept.

  4. After you read the material, practice your work.  On pp. 148 - 149, do problems # 21, 43, 44, 49, and 60.
    If you found these problems difficult or you made mistakes, do #      1 – 14 even, 22 – 24, 37 – 39, 51, and 53 - 55.
    If you got them all correct and found them easy, do #      48 – 50, 57 – 59, 61, and 66 - 69.

Assessment:  Complete pg. 149 # 70 – 74. 

Differentiation
Extensions:  Find the GCF of the following groups of numbers: 24, 136, 264  then  35, 147, 250  then 1450, 3600, 875 
Support (RtI tiers 2 & 3):  Review Lesson 3-3 and do # 1- 6 on p. 145 as well as # 75 – 80, p. 149
Do additional problems on pp. 148 - 149 as needed.
For extra practice, solve # 15 - 20 in the Extra Practice section on p. 686.
To further your understanding, read p. 150 “Reading a Math Lesson” and complete the work at the bottom of the page.

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Lesson 7: Simplifying Fractions     
Duration: One class period
Materials Needed:  Prentice Hall Mathematics 7, Course 2

Activities

  1. Equivalent fractions are fractions which have the same value or name the same amount.  ½ and 6/12 are equivalent fractions. You can write equivalent fractions by multiplying the numerator and denominator by the same number – a/a or by dividing the numerator and denominator by the same number – a/a. 

  2.                 1/3 · 4/4 = 4/12,   so 1/3 is equivalent to  4/12 

  3.                 9/24    3/3 = 3/8,   so 9/24    is equivalent to  3/8 

  4. Now play Fraction Frenzy at the LearningPlanet.com website.
  5. You should read Section 3-5, pp 151 – 152 in your text.  Use the Instant Check system to be sure you understand the material you read.  To do this, solve the problems beside the Check Understanding marks in each section, then check your answer with the key at the back of the book, beginning on p. 755.  If you miss a problem, reread the example to clarify the concept.
  6. After you read the material, practice your work.  On pp. 152 - 155, do problems # 7, 15, 23, 29, 34, and 43.
    If you found these problems difficult or you made mistakes, do #      2 – 24 even, 22 – 24, 26 – 29, 36, 41, 42, and 45.
    If you got them all correct and found them easy, do #      25, 31, 36, 38, and 44 - 50.

Assessment:  Complete pg. 155 Checkpoint Quiz 1, # 1 – 10.  

Differentiation
Extensions:  On p. 155, do # 51.  Now create 3 or more similar problems of equal or greater difficulty.  Provide answers on a separate sheet.
Support (RtI tiers 2 & 3):  Review Lesson 3-4 and do # 1- 7 on p. 151 as well as # 52 – 64, p. 155
Do additional problems on pp. 152 – 155  as needed.
For extra practice, solve # 21 – 26 in the Extra Practice section on p. 686.


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Lesson 8: Comparing and Ordering Fractions
 Duration: One class period
Materials Needed: Prentice Hall Mathematics 7, Course 2

Activities

  1. To compare and order fractions you need to find equivalent fractions with the same denominator.  You can use the least common denominator method described on pp. 156 – 157.  You can also find a common denominator by multiplying one fraction’s numerator and denominator by the denominator of the second.
    For example:    

    Compare     4/25 and 3/10. 
    The fastest way is to multiply:       4/25 · 10/10
     Then multiply: 3/10 · 25/25

     While 250 is not the LCD, it is a common denominator and you can easily compare the two fractions.   10/250  < 75/250. 
    Or, algebraically: 
              
    Compare     a/b and e/f

     Multiply: a/b · f/f  and e/f · b/b  Since they will have the same denominator, bf, you can now compare the fractions.

    Use whichever method works best for you!
     
  2. You should read Section 3-6, pp 156 – 157 in your text.  Use the Instant Check system to be sure you understand the material you read.  To do this, solve the problems beside the Check Understanding marks in each section, then check your answer with the key at the back of the book, beginning on p. 755.  If you miss a problem, reread the example to clarify the concept.
  3. After you read the material, practice your work.  On pp. 157 - 159, do problems # 11, 19, 26, 33, and 46.
    If you found these problems difficult or you made mistakes, do #      2 – 10 even, 22, 27 – 28, 35 – 38, 41, and 42.
    If you got them all correct and found them easy, do #      25, 26, 35 – 38, 41, 42, and 47 - 51.

Assessment:  Complete pg. 159 # 52 – 55. 

Differentiation
Extensions:  Make a poster which explains an alternative way to compare fractions.  Hint:  Look at your equivalency tables from previous lessons.
Support:  Review Lesson 3-4 and do # 1- 6 on p. 156 as well as # 56 – 64, p. 159
Do additional problems on pp. 157 – 159  as needed.
For extra practice, solve # 27 – 31 in the Extra Practice section on p. 686.

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Lesson 9: Solve a Simpler Problem and Look for a Pattern
Duration: One class period
Materials Needed: Prentice Hall Mathematics 7, Course 2 
 
11 Problem Solving Strategies Draw a Diagram Look for
a Pattern		 Make a
Graph		 Make an
Organized List		 Make a
Table
Simulate a Problem Solve a Simpler Problem Try, Check, and Revise Use Logical Reasoning Work Backward Write an Equation

Activities

  1. Here are some strategies for solving problems.  You might find it necessary to combine strategies, especially by solving a simpler problem first, then looking for a pattern.  Read the example carefully on pp. 160 – 161.
  2. You should read Section 3-7, pp 160 – 161 in your text.  Use the Instant Check system to be sure you understand the material you read.  To do this, solve the problems beside the Check Understanding marks in each section, then check your answer with the key at the back of the book, beginning on p. 755.  If you miss a problem, reread the example to clarify the concept.
  3. After you read the material, practice your work.  On pp. 161 - 162, do problems # 1 - 12.

Assessment:  Complete pg. 163 # 15 – 18. 

Differentiation
Extensions:  p. 162 # 13 and 14.
Support:  Review Lesson 3-1 and do # 1- 6 on p. 160 as well as # 19 - 32, p. 163
For extra practice, solve # 32 in the Extra Practice section on p. 686.
For practice with factoring, play the Factor Card game on p. 163.  You will need index cards to play the game
Supplementary Resources: http://www.learningplanet.com/sam/ff/index.asp

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Lesson 10: Mixed Numbers and Improper Fractions  
Duration:  One class period
Materials Needed: Prentice Hall Mathematics 7, Course 2 

Activities

  1. A mixed number is a number with a whole number and a fraction.  Its value is the sum of the whole number and fraction.  1½  is a mixed number.
    An improper fraction is a fraction where the value of the numerator is greater than that of the denominator.  5/3 is an improper fraction.
    To change a mixed number into an improper fraction, you multiple the whole number by the denominator of the fraction over itself, then add the numerators of the two:
    3    ½    = 3  + ½  which = (3/1 · 2/2)  +  ½  which = 6/2 + ½  which = 7/2
  2. Read Section 3-8, pp 164 – 165 in your text.  Use the Instant Check system to be sure you understand the material you read.  To do this, solve the problems beside the Check Understanding marks in each section, then check your answer with the key at the back of the book, beginning on p. 755.  If you miss a problem, reread the example to clarify the concept.
  3. After you read the material, practice your work.  On pp. 166 - 167, do problems # 38, 39, 42, 47, and 48.
    If you found these problems difficult or you made mistakes, do #      11 – 15, 21 – 25, 27, 28, 29 – 31, and 43 – 44.
    If you got them all correct and found them easy, do #      35 – 37, 44 – 46, and 50 - 52.  

Assessment:  Complete pg. 167 # 54 – 57.   
Differentiation
Extensions:  Make a list of at least 15 mixed numbers and/or improper fractions from real world situations.  Why might someone choose to use those rather than proper fractions? 
Support:  Review Lesson 3-5 and do # 1- 6 on p. 164 as well as # 58 – 67, p. 167.
Do additional problems on pp. 152 – 155  as needed.
For extra practice, solve # 33 - 38 in the Extra Practice section on p. 686.
Supplementary Resources:  http://www.learningplanet.com/sam/ff/index.asp

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Lesson 11: Fractions and Decimals
 Duration: One class period
Materials Needed: Prentice Hall Mathematics 7, Course 2

Activities
  1. First review the Investigation: Ordering Fractions on p 168.  Pay special attention to the relationship between fractions and decimal numbers.
    Decimals can be grouped into three different types: 

    Terminating - they end
    An example is 23.567

    Repeating:  one or more numbers to the right of the decimal point repeat in a pattern infinitely
    An example:  3.456456456    We show a repeating decimal by putting a bar over the numbers which repeat.

    Non-terminating, non-repeating:  there is no pattern to the numbers to the right of the decimal point and they appear to continue infinitely. 

    An example of a non-terminating, non-repeating decimal is     p (pi).

    To change a fraction to a decimal, divide the numerator by the denominator.  To change a decimal to a fraction, place the digits to the right of the decimal over the appropriate power of 10:  13.457 = 13     457/1000

    Here is that handy chart to help you remember some of the most common conversions:     

Fraction

1/10

1/8

1/5

¼

1/3

½

2/3

3/4

Decimal

0.10

0.125