Comprehensive Curriculum
Grade 7 Mathematics
Advanced Course
Cecil J. Picard
State Superintendent of Education
© September 2006
Grade 7
Advanced Mathematics
Table of Contents
Unit 1: Understanding Relationships 1
Unit 2: Rational Numbers 21
Unit 3: Algebraic Thinking 37
Unit 4: Patterns in Algebra 61
Unit 5: Geometry and Measurement 81
Unit 6: Measurement and Geometry 98
Unit 7: Data and Statistics 112
Unit 8: Understanding Probability 125
This course covers both Grade 7 and Grade 8 GLEs. It is designed for Grade 7 students intending to take Algebra I in Grade 8.
Grade 7
Advanced Mathematics
Unit 1: Understanding Relationships
Time Frame: Approximately four weeks
Unit Description:
This unit begins with the use of different problem-solving strategies. The relationship of fractions, decimals and percent will be explored in this unit. A deeper understanding of the relative size of numbers is emphasized. Place values are associated with powers of 10, and scientific notation is introduced as a way to rewrite large values. The unit also introduces operations with integers through the use of manipulatives.
Student Understandings
Students will review the different problems solving strategies as they begin this unit. Students will demonstrate their understanding of fraction, decimal, integer, and ratio/percent representations through comparing, ordering, contrasting and connecting these representations of rational numbers in problem-solving contexts. The students will use order of operations and proportions while exploring rational numbers. The students will demonstrate an understanding of integer operations by observing patterns and conjecturing about rules.
Guiding Questions
- Can students determine an appropriate problem-solving strategy?
- Can students represent in equivalent forms and evaluate fractions, percents, decimals, integers and ratios?
- Can students compare rational numbers using symbolic notation as well as use position on a number line?
- Can students recognize, interpret, and evaluate problem-solving contexts with fractions, decimals, integers, and ratios?
- Can students demonstrate the equality of ratios in a proportion?
- Can students set up and solve simple percent proportions?
- Can students determine and justify the reasonableness of answers to problems involving rational numbers?
- Can students explain and use integer operation rules?
Unit 1 Grade-Level Expectations (GLEs)
7th grade
Number and Number Relations
1. / Recognize and compute equivalent representations of fractions, decimals, and percents (i.e., halves, thirds, fourths, fifths, eighths, tenths, hundredths) (N-1-M)
2. / Compare positive fractions, decimals, percents, and integers using symbols (i.e., <, £, =, ³, >) and position on a number line (N-2-M)
3. / Solve order of operations problems involving grouping symbols and multiple operations (N-4-M)
6. / Set up and solve simple percent problems using various strategies, including mental math (N-5-M) (N-6-M) (N-8-M)
7. / Select and discuss appropriate operations and solve single- and multi-step, real-life problems involving positive fractions, percents, mixed numbers, decimals, and positive and negative integers (N-5-M) (N-3-M) (N-4-M)
8. / Determine the reasonableness of answers involving positive fractions and decimals by comparing them to estimates (N-6-M) (N-7-M)
9. / Determine when an estimate is sufficient and when an exact answer is needed in real-life problems using decimals and percents (N-7-M) (N-5-M)
10. / Determine and apply rates and ratios (N-8-M)
11. / Use proportions involving whole numbers to solve real-life problems (N-8-M)
12. / Evaluate algebraic expressions containing exponents (especially 2 and 3) and square roots, using substitution (A-1-M)
Data Analysis, Discrete Math and Probability
35. / Use informal thinking procedures of elementary logic involving if/then statements
(D-3-M)
8th grade
Number and Number Relations
1. / Compare rational numbers using symbols (i.e., <, £ , =, ³, >) and position on a number line (N-1-M) (N-2-M)
2. / Use whole number exponents (0-3) in problem-solving contexts (N-1-M) (N-5-M)
3. / Estimate the answer to an operation involving rational numbers based on the original numbers (N-2-M) (N-6-M)
4. / Read and write numbers in scientific notation with positive exponents (N-3-M)
5. / Simplify expressions involving operations on integers, grouping symbols, and whole number exponents using order of operations (N-4-M)
6. / Identify missing information or suggest a strategy for solving a real-life, rational-number problem (N-5-M)
7. / Use proportional reasoning to model and solve real-life problems (N-8-M)
8. / Solve real-life problems involving percentages, including percentages less than 1 or greater than 100 (N-8-M) (N-5-M)
9. / Find unit/cost rates and apply them in real-life problems (N-8-M) (N-5-M) (A-5-M)
Sample Activities
Activity 1: Missing! (GLEs: 7th – 7; 8th – 6)
It is important for the students to establish problem-solving strategies early in the year. This activity was written to help students recognize the steps in problem solving. Lead a class discussion about problem-solving strategies that the students have used. Have students make a list of basic steps involved in problem solving: a) understand the problem; b) make a plan, sketch or diagram of the problem; c) carry out the plan (do the computation); and d) determine that the solution makes sense. Discuss the different problem-solving strategies such as: a) working backwards; b) sketching models or diagrams; c) making tables, charts or graphs; or, d) setting up and solving a simpler problem first.
Put a situation like one of those listed below on the overhead and have students write a plan for solving the problem. Give the students time to solve it and then have students discuss solution strategies. Ask, Does anyone have a different method that was used to solve the problem? Discuss methods and make sure students verbally explain why their method worked for them.
· Samantha is floating on a raft 75 feet from the shore at 10 a.m. Every hour she moves herself forward 15 feet with her arms, but she stops when she gets tired and the current pulls her backward 6 feet. At this rate, what time will Samantha reach the shore? Explain your solution.
Solution: 6:20 p.m. Since she gets about 9 feet every hour, 75 ¸ 9 = 8 hours, but she will have three feet left. If there are 3 feet left and she moves 9 ft in an hour, then she will need 1/3 of an hour to get to shore. This is 20 minutes.
· There are six boys in a race. Carl is ahead of Bill, who is two places behind Frank. Allen is two places ahead of Dwight, who is two places ahead of Evan. Evan is last. Which of the boys is closest to the finish line? Explain your solution.
Solution: Carl is closest to the finish line. Carl is first, Allen second (two places ahead of Dwight), Frank is third (two places ahead of Bill), Dwight is fourth, Bill is fifth (two people behind Frank), and Evan last (two places behind Dwight).
· A group of students has gathered around the center circle of the basketball court. The students are evenly spaced around the circle. Student #11 is directly across from student #27. How many students have gathered around the circle? Explain your solution.
Solution: 32 students. Sketch a model to get an idea as to positioning. Making a table of the opposites as they draw a diagram will give the students the answer. Example: We know 11 is across from 27, which means that students 12 through 26 must sit between them. Number 19 is in the middle position in a listing of these fifteen numbers (median). This means that the number opposite 19 would have to be the 8th number away from 11, which is 3. 3 would also have to be the 8th number away from 27. The numbers between 3 and 27 would be 1, 2, 28, 29, 30, 31 and 32. The answer is 32 people at the table.
Students also have to be able to determine when there is not enough information available to solve problems and locate the additional information needed to solve a problem. They also need practice in devising problems. Have the students examine problems like the ones below and discuss how these are different from the earlier problems. (They are missing information.) Have them work in pairs or small groups to determine the missing information and find the solution.
· The world record high dive is 176 feet 10 inches. What is the difference between Jack’s highest dive and the world record?
· Mary wants to find the amount of carpet needed to carpet her bedroom. She measures the length of the room. How much carpet does she need to carpet the bedroom?
· Greg Louganis holds 17 U. S. national diving records. How many of these did he earn before the 1988 Olympics?
Activity 2: Grouping Dilemma! (GLEs: 7th – 3; 8th – 5)
Display the tile pattern at the left on the overhead either with tiles or a sketch. Give the students directions to find the total number of tiles without counting each one. Have the students sketch the pattern and draw circles or “loops” around groups of tiles to help them determine the total number of tiles. It is important that the students explain how their grouping method matches their mathematical expression or equation.
Examples: (There are many more groupings.)
.
or some students even see
Allow time for as many different groupings as the students develop, asking questions that help them justify how their groupings matches their mathematical expression. Next, ask students to use information from the classroom discussions to determine how many square tiles would be needed if tiles were arranged in this same manner with the top right tile missing, but there were 8 rows and 8 columns. Lead discussion as the students determine which of the methods used earlier make it possible to find the number of tiles. (63) Extend the problem by asking students how many rows and columns would be in an arrangement using the pattern of the missing corner tile that has a total of 120 tiles. (11 columns, 11 rows, with the corner missing)
Activity 3: Decimal Comparisons - Where’s the Best Place? (GLEs: 7th – 2; 8th –1, 2)
Discuss place value and what the decimal part of a number signifies. Make sure the students understand that the part of the number to the right of the decimal point shows the fractional part of a whole. Start with an understanding that the first place to the left of the decimal point is the ones place and the value of that number in that position is determined by multiplying the digit by 1. Ask students for the value of the number in the next position to the left and continue asking questions that lead the students to see that each place to the left of the ones place is simply the next power of 10. Similarly, lead the students in a discussion by looking at the value in the tenths place as some number multiplied by or . Give students time to establish the powers of ten for each place from the thousandths to the thousands place. Once the students have established the power of ten that represents each place value, introduce the idea of 10o =1 so they can see that every place value is some power of 10.
Place the students in groups of 4 to play the game, Where’s the Best Place? The object of the game is to build the greatest number possible. These are the rules for the game:
§ Give each player a number card sheet which has several game cards and have groups play the game several times. There is a number card sheet sample after the activity pages.
Example of Game Card 1
§ Have students shuffle ten cards numbered 0 through 9 and place them face down in a pile.
§ One player draws a digit card from the pile. Each player must decide privately where he/she wants to write that digit on his/her game card.
§ After the player writes a digit on the card, he/she cannot erase it and place it somewhere else. Once a digit is drawn, it cannot be used again in that game.
§ The game is over when all places on each game card are filled.
§ Write an inequality using the four numbers generated by the group.
§ The player with the greatest number wins.
After the class has played the game, have the students discuss some of the strategies they used as when trying to form the largest number possible. This might be easier for the students if they are asked where they would place a 7 if they got that number on the first draw and have them justify their reasoning.
Activity 4: Fraction Comparisons (GLEs: 7th – 1, 2; 8th – 1)
To check the depth of understanding in dealing with equivalent fractions, have the students complete the pizza problem below while working with a partner or in a group of 4. Circulate around the room and ask questions to find what strategies the students are using to find equivalent fractions.
Provide chart paper for the students to complete their work. Students should be prepared to share their thinking with other groups or the class in about 20 minutes.
Dana, Susan and Bill ordered a pizza. Each would receive the same amount, but Dana wanted her portion divided into 3 pieces, Susan wanted her portion divided in 2 pieces and Bill wanted his portion to be one large piece.
1. Sketch the pizza and label each student’s pieces with his/her name and the fraction that represents the size of each piece as related to the whole. Use mathematics to show how the number of pieces each student gets forms a fraction that is equivalent to the other students’ portions. Example: If one student asked for his part of the pizza to be 4 parts, this would be of the whole and would equal if another student wanted his part divided into 6 parts.