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7th GradeMath 6061

Jeffery Ostrom

Number Sense

Executive summary

During this unit student will gain a greater understanding of the 7th grade Minnesota Number & Operation standards, through active activities done in groups and individual work. We will be covering standards: 7.1 Number & Operation, Read, write, represent and compare positive and negative rational numbers, expressed as integers, fractions and decimals; 7.2 Number & Operation Calculate with positive and negative rational numbers, and rational numbers with whole number exponents, to solve real-world and mathematical problems. We will be using a variety of methods to learn these standards. We will be using graphs, tables, manipulatives (concrete), verbal and formulas, to help better understand these math standards. There will be a pre and post-test given to assess how the students have progressed in their interaction of the unit. The assessment questions will come from activities done in class, MCA Sample test, and other sources. Then we will work through each activities building on prior skill and learning new ones that will help students better understand the standards that they will need to cover in 7th grade and on. They will be able to solve problems like:

Starting with the first activity “Zip, Zilch, Zero” student will be getting a hands on approach to work with integers. You will then move into “Order of Operations Bingo”, in this wonderful activity student will practice using, order of operations. We will then work on rational numbers and proportions. In this lesson, students use real-world models to develop an understanding of fractions, decimals, unit rates, proportions, and problem solving. The three activities in this investigation center on situations involving rational numbers and proportions that students encounter at a bakery. In the lesson, Happy Birthday to You, students look at patterns in calendars to determine the day of the week for a particular date. Students also use division to explain why those patterns occur, and then relate them to modular arithmetic. The final assessment is for students to find the day of their birth. Big Math and Fries is the next lesson. Here students will perform calculations, including percents and conversions. The next few activities will be spent on having students develop number sense through a series of three hands-on activities. Students explore the following concepts: the magnitude of a million, fractions between 0 and 1, and the effect of decimal operations. Lastly students will work through a credit card scenario with a teaser rate, minimum payments, fees, and rate increases for being late, in a activity called, “The cost of Being Late”.

Table of Contents

(1 day) Pre-Test

(1 day) Zip, Zilch, Zero

(1 day) Order of Operations Bingo

(3 days) Understanding Rational Numbers and Proportions

(1 day) Happy Birthday to You

(1-2 days) Big Math and Fries

(3 days) Too Big or Too Small

(2 days) The Cost of Being Late

(1day) Post-Test

Pre-Test Number Sense Name_____________________________

Four points graphed on a line. Which point is located opposite of -2?

Is this statement true or false? Justify your answer.

Which shows a model of -3 + 4?

Simplify. 2 x (-5) + 4(2 + 1)2

Which is equivalent to 52

15?

Jan’s starting weight was 220 after the first week her weight was 212 pounds. Mark’s starting weight was 432 after the first week his weight was 421 pounds. Who was the biggest loser for that week?

Simplify. 8 – 2(n + 4)(-3)2

In this equation, which describes n? n = 1 ÷ 17

Find the cost of various-sized pieces, 12

, 14 &

18 given that a whole cake costs $12.72?

Keiko bought fruit for $0.59 per pound. She spent $3.00. About how many pounds of fruit did she buy?

Which shows the numbers ordered from least to greatest?

A car’s original price was $26,500. Mr. Thomas paid $23,585. What percent of discount did Mr. Thomas receive on the car?

Zip, Zilch, Zero

A positive and negative number become more than marks on paper when students play this variation of the card game, Rummy. Engaged in a game involving both strategy and luck, students build understanding of additive inverses, adding integers, and absolute value.

Learning Objectives Students will:

Develop a better understanding of how positive and negative integers relate to each other

Investigate ways to combine integers to get 0 Become more fluent in addition of integers

Materials decks of regular playing cards Zip, Zilch, Zero Rules and Record Activity Sheet Instructional Plan

To begin the lesson, review adding integers. Show examples that include both positive and negative integers.

LaunchThen divide the class into groups of three or four. Students will compete individually against the other people in their group. Distribute the Zip, Zilch, Zero Rules and Record, and read the rules aloud to students. As necessary, answer questions and provide additional explanation. For example, you might want to show the example below to give students an understanding of how to make a Zip.

It will be helpful to couple this example with the mathematical equation that it represents:

(-7) + 9 + (-2) = 0

This will help students to see that red cards represent negative numbers, as well as an example of how to make zero using more than just additive inverses. When you are certain that all students understand the rules of the game, tell them that they will play a practice hand before keeping score, to make sure that everyone understands the rules of the game.

Zip, Zilch, Zero Rules and Record Activity Sheet

ExploreDistribute the Zip, Zilch, Zero activity sheet. Give all instructions before giving cards to the groups so they can focus on the preparation. Point out that Question 1 concerns the practice hand they will play. For this hand, when someone goes out, everyone should show their remaining cards. They will then estimate who will end up with the highest score and write their predictions on the paper. This allows students a window into each others' thinking and gives them practice with estimation and integers. They will also accurately determine scores and reflect on the quality of the prediction.

Hand out one deck of playing cards to each group. Monitor groups closely during the practice hand. Students may think they have to match a red and black number exactly to make a zip. Remind them that they will get rid of their cards faster if they can make combinations with more than two cards. In addition, students may be familiar with games where all face cards are worth 10; in this game, J = 11, Q = 12, and K = 13, and it may be helpful to remind students of this rule several time.

Groups work through the activity sheet as they play. The activity sheet requires students to verify each other's scores to get more practice and to prevent disagreements later. If time is short, you can reduce the number of hands required for a game.

Share Have students share their best strategies with the class and explain why they worked well.

Questions for Students How can you make the most points when you lay down a zip? [Use cards with high point values, and use a lot of cards with each zip.] What will happen if you get down to only one card in your hand? [You will never be able to go out because you have to discard to go

out, and you have to draw at least one card on every turn.]

When is it a good idea to pick up more than one card from the discard pile?

[You should pick them up if you can make a lot of points from using several cards in the pile. Another time to pick them up is when you only have one card in your hand and need more cards to be able to go out.]

When is it not a good idea to pick up more than one card from the discard pile?

[You should not do it if it looks like someone is about to go out because you could get caught with a lot of extra points in your hand.]

Why might you want to hold cards in your hand even if you could lay them down in a zip?

[If playing the zip gets you down to just one card, you limit your options for the rest of the game. If you have a sense of the dramatic, you might like to play all your cards at once to go out, but you take a risk of someone going out before you've played your zips.]

What other situations can be thought of in terms of opposites combining?

[Answers will vary. Samples: debts and earnings, driving one direction and then back the other direction, digging a hole and filling it up, balancing chemical formulas and equations]

List the ways you could play cards to balance out a black 2 and a black 4 in one zip.

[There are 10 ways, summarized in the table below.]

1 CARD 2 CARDS 3 CARDS 4 CARDS 5 CARDS1 red six 1 red ace

1 red five2 red aces1 red four

3 red aces 1 red three

4 red aces1 red two

1 red two1 red four

1 red ace1 red two1 red three

2 red aces2 red twos

2 red threes 3 red twos

Assessment Options Circulate around the classroom as students play the game. Listen for

evidence that they understand the math concepts or have developed a good strategy. Record that evidence.

Have students fold a piece of paper in half. Have them draw a big plus sign on one side and a big minus sign on the other side. Write different integer addition problems on the board, one at a time. Have students hold up the side of the paper that shows what the sign of the answer would be.

Deal a sample hand and challenge students to make a given number other than zero using the maximum cards possible.

Extensions1. If you would like to play this game periodically and/or make a tournament of

it, you can use the Zip, Zilch, Zero Record activity sheet to record the games.2. Play the game the same way except that books must equal 24 using any

operations. For this game, students should write down the expression that creates their book and have it verified by another student before it can be counted. For example, if a student plays a black 10, a red 2, and a black 3, they must write on their paper "3(10 + -2)."

3. Play the game the same way but randomly pick a value for the zips (now called "melds" because they do not have to equal zero any more). The target could be the first card turned over after cards are dealt to players. The game could also be progressive with the first hand's target being 1; the second hand's target being -2, then 3, then -4, etc.

4. Play "Human Integers." Split the class randomly into two groups, forming two single file lines. Give matching party hats to each student in one of the lines, and different matching hats to students in the other line. One line is positives, and one line is negatives. Give students an integer problem to model. For example, 5 + (-3) = 2. If black hats are positives and red hats are negatives, the first 5 students with black hats would go to the front of the class along with the first 3 students with red hats. The students make 0s by having a black hat stand next to a red hat. The two black hats without a partner represent the sum. The participants go to the end of the line.

5. For practice on subtracting integers, deal out 6 cards to each player. Turn the next card face up. This is the "target." Players must use 2 cards in their hands to make a subtraction problem with the target number as the difference. For example, if the target is a black king (13), a student could play a black 8 and a red 5 because 8 – (-5) = 13. If a player cannot make the target number, they do not lay any cards down and miss the opportunity to score on this round. Once all members of the group have created their problems or passed, deal 2 new cards to each player (including the people who may not have been able to play their cards) and turn over a new target number. After 4 rounds, determine scores by counting the cards played as positives and the cards remaining in a player's hand as negatives.

6. Have students analyze different scoring methods. For example if the scoring method was to add up the cards played along the ones remaining in your hand with blacks and reds still representing positives and negatives respectively before subtracting what you have remaining in your hand from the value of the cards you played. How do you think the change would affect scores? Why do you think so?

[Answers can vary. Sample: The cards played should all add up to zero, so you would be subtracting the value of the remaining cards from zero. If that is a positive value, your score for the hand is negative; and vice versa. Or whether or not scores increase or decrease depends on what the scores were under the original rules and how the hand played out under the new rules. If a person had

negative scores under the regular rules, and their hand had an abundance of red cards remaining, under the new rules the scores would considerably improve. On the other hand, if the opposite were true, their scores would decline.]

Summarize1. Develop and analyze algorithms for computing with integers and develop

fluency in their use.2. Understand the meaning and effects of arithmetic operations with integers.3. Use the associative and commutative properties of addition to simplify

computations with integers.

Teacher Reflection How did students demonstrate that they understood additive inverses

and/or integer addition? Did you allot an appropriate amount of time for the lesson? Did students tire

of the game before the period was over? Was there enough time for students to finish a game of 4 rounds? What, if anything, would you change about the time allotted?

Did students behave appropriately for a classroom game situation? If yes, what made it so; and if no, how can you help students make better choices next time?

Did you make adjustments to the lesson? Were they effective? Why or why not?

NCTM Standards and Expectations Number & Operations 6-8

1. Develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use.

2. Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers.

3. Use the associative and commutative properties of addition and multiplication and the distributive property of multiplication over addition to simplify computations with integers, fractions, and decimals.

This lesson was prepared by Kim Clary as part of the Illuminations Summer Institute.

Zip, Zilch, Zero Rules NAME_________________________

Setup: The game is for 3 or 4 players. Use a standard 52-card deck. Choose a dealer by drawing for the high card. Shuffle the deck and deal 7 cards to each player. With the remaining cards, form a draw pile in the middle of the table. Place the top card face up next to the draw pile to start a discard pile.

Gameplay: Each turn, a player must follow this sequence. 1. Draw one card, either from the top of the draw pile or the top of the discard pile.

A player may choose to draw more than the top card from the discard pile but must take all the cards on top of it and must play that bottom card in a zip during that turn.

2. Play any zips. (See “Zips” below)3. Discard one card, adding it (face up) to the top of the discard pile. The card

should be placed so that the cards below can still be seen. The previous discards are still available for play.

If a player chooses to draw the top card on the discard pile in step 1, that card may not be discarded during the same turn in step 3.

Zips consist of at least two cards from a player’s hand that sum to 0. Black cards count as positive numbers and red cards count as negative numbers. Aces are worth one point, number cards are worth face value, jacks are worth 11, queens are worth 12, and kings are worth 13.

Example 1: A red 6 and a black6 Example 2: A black king, a red 2, a red 3, and a red 8

Going Out: A player goes out when the last card in his or her hand is played as a discard.

Scoring: When a player goes out, the hand is scored. Players subtract the absolute value of any cards remaining in their hands from the absolute value of cards they played in zips.

SCORING EXAMPLES

ZIPPED CARDS

REMAINING CARDS

VALUE OF ZIPPED CARDS

VALUE OF REMAING

CARDSSCORE

Sample Hand 1

Red 6, black 6, black king, red 2, red 3,

red 8

Black 7, red 3 38 10 28

Sample Hand 2

Red 7, black 2, black 5, red

3, black 2, black ace

Red king, black queen 20 25 -5

Winning: The winner is the player with the most points after four hands are played. © 2010 National Council of Teachers of Mathematics http://illuminations.nctm.org

Date opponents My score My order of finished

Order of Operations Bingo

Instead of calling numbers to play Bingo, you call (and write) expressions to be evaluated for the numbers on the Bingo cards. The operations in this lesson are addition, subtraction, multiplication, and division. None of the expressions contain exponents.


Evaluate expressions using the order of operations on +, –, ×, and ÷ Use mental arithmetic to evaluate expressions.

Materials Order of Ops Bingo Sheet Bowl, jar, or hat Chips for marking spaces on the Bingo cards

LaunchStudents can often rattle off the acronym PEMDAS or "Please Excuse My Dear Aunt Sally" as being associated with the order of operations. Putting this memory into practice can be more of a challenge. By practicing the correct order with a motivating game of Bingo, students will be more eager to be accurate in their calculations.

One misconception by students is that all multiplication should happen before all division because the multiplication comes before division in the acronym. In fact, multiplication and division have the same precedence and should be evaluated as they appear from left to right.

Incorrect Correct

Similarly, addition comes before subtraction in the acronym, yet they have the same precedence.

Incorrect Correct

ParenthesesExponentsMultiplication / DivisionAddition / Subtraction

Try giving students an additional example before starting the game.

Playing Order of Operations Bingo

To prepare the materials for the game, you will need to print the Order of Ops Bingo Sheet. The first two pages contain 50 expression strips, which you will need to cut out and place in a bowl, jar, or hat. The third page contains two bingo cards; you will need to photocopy this sheet, cut the copies in half, and distribute a sheet to each student.

Order of Ops Bingo Sheet

The object of the game is to get five numbers in a row, vertically, horizontally, or diagonally, just as in the regular game of bingo.

NOTE: The operations used for this lesson are addition, subtraction, multiplication, and division. None of the expressions contain exponents or parentheses.

ExploreDistribute a Bingo card to each student before starting the game. Give students the following instructions:

Choose one space on the board as the "free" space and write the word FREE.

Choose numbers to write into the other 24 boxes on your Bingo card. Make sure you choose numbers in the ranges given at the top of each column. That is, numbers in the first column ("B") must be in the

range 1-10, numbers in the second column ("I") must be in the range 11-20, and so on. [This ensures better distribution of the numbers.]

You are not allowed to repeat any numbers.

Place all of the expression strips in a bowl, jar, or hat, and choose them one at a time. After each selection, write the expression on the board or overhead so students can evaluate it. Students should copy down and evaluate the expression on their own paper. For the first few turns, you may want to model how the numerical value is determined for the expression by writing in any applicable parentheses and going through the steps of evaluation. Make sure you write out the steps, just as you'd like to see the students do themselves. Once the number is determined, students can look for the number on their Bingo card and mark it with a pencil or a chip.

The value (i.e., the "answer") for each expression follows the expression on each strip, so be sure to share only the expression, saving the answer to verify a winner.

Keep picking expressions. Students should calculate the value for each expression, and then mark the square with that number on their card (if that number appears on their card, of course). When a student believes that she has correctly completed a column, row or diagonal on her card, she should yell, "Bingo!"

When the game has a potential winner, ask the student to call out the numbers that make the winning row, column, or diagonal. With the class, determine if the numbers that the winning student calls are indeed values from expressions that have been called out to check the math and verify the win.

To extend the game for another winner, change the rules to require 2 runs of 5 chips, or framing the exterior square of the board (16 pieces).

If students use chips instead of crossing off numbers with a pen or pencil, then they can exchange cards and play again. In order to start a second or subsequent game, all expressions used in the previous game are returned to the bowl, jar, or hat for a fresh start.

ShareQuestions for Students

1. The order of operations says to multiply and divide first. What does this mean?

2. [It means that multiplication and division are performed before addition or subtraction. However, it does not mean that multiplication should be done before division. Multiplication and division have the same precedence, so if either multiplication or division occur with in an expression, perform these operations from left to right.]

3. What does it mean for addition and subtraction to have the same precedence?

4. [It means that addition should not be done before subtraction. If either addition or subtraction occur with in an expression, perform these operations from left to right.]

5. In the expression 3 + 4 × 5 – (3 + 2), explain the order in which the operations should be performed, and evaluate the expression.

6. [Operations within parentheses are done first, so add 3 + 2 = 5. This changes the expression to 3 + 4 × 5 – 5. Then, multiplication (and division, too, though there's none in this expression) are performed before addition and subtraction, so multiply 4 × 5 = 20. The expression is now reduced to 3 + 20 – 5. Finally, perform the addition and subtraction left to right to give 18.]

Assessment Options1. Have students evaluate several expressions that contain several operation

symbols.2. Give students a list of numbers with no operation symbols, and ask them to

place the symbols so that a specific result occurs.a. Example: Given the list of numbers 1 2 3 4 5, can you write in the

symbols +, –, × and ÷ so that the value of the expression equals 8? Any of the symbols may be used more than once and all of the symbols don’t have to be used.Answer: 1 + 2 × 3 – 4 + 5

3. Ask students to create some expressions of their own. In pairs or groups, students evaluate each other's expressions and see if there is agreement on the value of each expression. Note that students may agree on an incorrect value due to a misconception in the order of operations.

Extensions1. Create and evaluate expressions that are more complex.2. Present expressions containing exponents or nested parentheses (if students

have had exposure to these concepts and their notation)

3. Ask the class to create expressions whose values are whole number from 0 through 75. This time, the columns on the Bingo cards have a range of 15: 1-15, 16-30, 31-45, 46-60, and 61-75.

Summarize1. Develop and use strategies to estimate the results of rational-number

computations and judge the reasonableness of the results.2. Understand the meaning and effects of arithmetic operations with integers.

Teacher Reflection What are the most common misconceptions students have regarding the

order of operations? What can be done to break those misconceptions? What examples were most helpful in getting students to understand the

order of operations? What other examples would help students to better understand the order of operations?


3. Develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results.

4. Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers.

This lesson was developed by Zoe Silver.

Understanding Rational Numbers and Proportions

In this lesson, students use real-world models to develop an understanding of fractions, decimals, unit rates, proportions, and problem solving.

The three activities in this investigation center on situations involving rational numbers and proportions that students encounter at a bakery. These activities involve several important concepts of rational numbers and proportions, including partitioning a unit into equal parts, the quotient interpretation of fractions, the area model of fractions, determining fractional parts of a unit not cut into equal-sized pieces, equivalence, unit prices, and multiplication of fractions.


represent parts of a whole using an area interpretation of fractions determine the fractional part of a whole when parts are not cut into equal-

sized pieces develop an understanding of the quotient interpretation of fractions find the unit cost of items that are part of a set determine the relationship among parts of a whole that are unequal-sized

pieces express fractional parts of a whole as decimal equivalents

Materials Making Four Pieces Overhead Cakes Cut Into Eighths Activity Sheet Cakes Cut Into Fourths Activity Sheet Scissors Calculators (optional)

Day1 Activity 1: Customers Cut the Cake

LaunchEach day the local baker makes several rectangular sheet cakes, which he cuts into eighths. He sells 1/8 of a sheet cake for $1.59. As part of a new promotional campaign for his store, he wants to cut his sheet cakes into eighths a different way each day. Customers who suggest a new way to cut the cakes into eighths win a free piece of cake each day for a week. What are some of the different ways to cut the cake?

Some questions to ask students include:

What is the shape of the baker's cakes? [Rectangular cakes.] What are the restrictions on the ways the cakes can be cut? [Pieces must

contain the same amount of cake; they do not have to be the same shape.] How can we verify that pieces that are not the same shape contain the same

amount of cake? [Cut the pieces into smaller parts and lay one on top of the other.]

This activity gives students opportunities to represent parts of a whole by using an area model of fractions. Many students know that fractions often refer to equal-sized parts of a unit, but they frequently overgeneralize and believe that the pieces have to be congruent rather than merely contain the same area.

ExploreEncourage students to solve the problem in pairs or in small groups by using the Cakes Cut Into Eighths activity sheet.

Alternatively, students can use rectangular pieces of paper to model the cakes, sketching the shapes or cutting the paper into eight pieces and verifying the equivalence of the pieces by cutting and overlapping. Some students may have a limited view and think that all the cuts must be parallel to one side of the rectangle. Challenge them to think of other ways to make the cuts. (It is important for students to know that each rectangle on the activity sheet represents a full cake.)

Share Have students place their designs on the chalkboard or the overhead projector. Ask the students to decide which designs are the same and which are different. Examples of some diagrams are shown below. Have the students discuss whether the two rectangles in this figure are cut differently.

What factors should be considered when deciding whether the two designs are different? (The number of pieces, the equivalence of the pieces, and whether the location of the pieces makes a difference should be considered.) The two rectangles in the above figure contain the same eight pieces, but the pieces are arranged differently. The students can decide whether they want to consider these as two different arrangements.

Another way that must be considered is using cuts that are curves or combinations of line segments, such as the examples in the figure below.

The equivalence of shapes formed by cuts that are curves is difficult to determine but is a good investigation in itself. The equivalence of shapes that are formed by cuts that are combinations of straight line segments is easier to determine. Including these types of shapes, however, again greatly increases the number of possibilities. By interacting with students, decide which designs to include in the final count.

Ask each group to choose one design and either to post it on the chalkboard or bulletin board or to draw it on the overhead projector. Students in each group should be prepared to explain how they know that their method shows eighths. One way to verify that a solution does in fact result in eighths is to cut up the individual pieces further and lay them on top of each other to verify the equivalence of their areas.

As a follow-up activity, teachers may choose to discuss with students why or why not each cake on below is cut into eighths.

Day 2 Activity 2: You Can Eat Your Cake and Have It, Too!

LaunchThe baker is conducting a second contest, this time for his employees. As part of a new promotional campaign for his store, each day he wants to feature sheet cakes that have been cut into four pieces in a different way. The pieces do not have to be equal for this promotion. The baker has challenged his employees to suggest interesting ways to cut the cakes into four pieces. The employees must also determine the price for each piece. The bakery sells 1/8 of a sheet cake for $1.59. What are some of the different ways the cakes can be cut, and how much should each piece cost?

ExploreSome questions for the students to discuss include:

1. What are the restrictions on the ways the cakes are cut? [Each cake must be cut into four pieces, not necessarily equal-sized pieces.]

2. How can we determine the fractional parts of the pieces we cut? [We can use equivalences we know, such as 2/4 is equivalent to 1/2, to find the value of each part, if we partition by finding parts of parts.]

3. What will happen if we just cut four pieces at random? [It will be difficult to determine the size of each piece.]

Instead of focusing on making equal-sized pieces as in the previous activity, this activity explores determining the fractional parts and cost of pieces when a unit is cut into four unequal parts.

Encourage students to solve the problem in pairs or small groups by using the Cakes Cut Into Fourths activity sheet or rectangular paper to model the cakes. They sketch the pieces or cut the rectangles into four pieces and determine the value of each piece.

Cakes Cut Into Fourths Activity Sheet

One way to find the value of each piece is to add partitioning lines so that the whole is partitioned into equal-sized pieces. Students may remember some of the ways they cut the cakes into eighths in the first activity, which may help them. Once again, weighing could solve this problem.

The following overhead can be projected after students have had time to create their designs.

Making Four Pieces Overhead

ShareHave students share and discuss their designs. The rectangles shown below show a few possible ways to cut the rectangular cakes into four parts. The pieces have been labeled to show the fraction of the cake they represent.

Students must then find the cost for each piece, if 1/8 of a cake costs $1.59. What should the total cost of one whole cake be? (If 1/8 of a cake costs $1.59, then a whole cake should cost eight times as much, or $12.72.)

Challenge students to explain and verify their solutions.

Day 3 Activity 3: That's the Way the Cookie Crumbles!

LaunchYou bought a baker's dozen (13) of cookies that you want to share equally with your family. How many cookies will each person get?

ExploreAsk students to compare this problem with the one posed in the first activity. Give them time to think about the similarities and differences between this problem and the problem posed in Activity 1.

[This problem is similar to the first problem, that of cutting rectangular sheet cakes into eighths in different ways, because they both involve partitioning a unit into parts. This problem is different in several ways: The whole or unit in this group of 13 cookies; the problem does not specify exactly into how many pieces to cut each cookie or how many people are sharing the cookies; the problem has different solutions for students who have different-sized families; and this problem involves a different interpretation of rational numbers — the quotient interpretation. The quotient interpretation refers to the fact that in this problem, in which thirteen cookies are being shared by n people (n is the number of people in the family), the number of cookies each person receives is the quotient, 13 ÷ n.]

Consider grouping students according to the number of people in their families. Students can draw 13 circles on a piece of paper to practice dividing the cookies. After each group has solved the problem, share the solution processes with the whole class. Have each student complete a table similar to the sample below.

The table should include all the different-sized families of students in the class and also contain a few other family sizes, including one or two families that are larger than the largest family in the class. Students should begin completing the table by including solutions from the groups in the class and then working on solving the problems for other family sizes.

ShareDiscuss the solutions, focusing on the patterns students see in the number of cookies for each person as the size of the family changes. The goal for students is to generalize that the number of cookies for each person is equal to the number of cookies divided by the number of people sharing them.

Ask students to write a rule that represents the number of cookies each person receives if thirteen cookies are shared by n people (where n is the number of people in the family).

[The number of cookies each person receives is the quotient, 13 ÷ n.]

Ask students to describe the way they solved the problem in their groups (e.g., by drawing a picture, using long division, and so on) so that a student in another class would understand what problem was solved and how it was solved.

Summarize Understand and use the inverse relationships of addition and subtraction,

multiplication and division, and squaring and finding square roots to simplify computations and solve problems.

Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers.

Assessment OptionsAssess student understanding of naming fractions and fractional equivalents by focusing on how students solve this problem. For example, are students able to recognize that 4/8 is equivalent to 1/2? Do students use their knowledge that 6/8 is equal to 6 one-eighths to help them find the cost of 6/8 of a cake (i.e., since 1/8 of a cake costs $1.59, then 6/8 of a cake costs 6 ×:1.59 = $9.54)?

Extensions Students find the cost of various-sized pieces, given that 1/8 of a cake costs $1.59 and a whole cake costs $12.72. The following table is a sample. (Tables may include other fractional parts and need not be limited to eighths, fourths, and halves.) Students may wish to use calculators with fraction capability to help them find the various prices. Using calculators may help students focus on the reasonableness of their solutions rather than on the calculations.

How many cookies would each person get if... three people shared twenty cookies? [20/3, or 6 and 2/3 cookies for each

person] eight people shared twenty cookies? [20/8, or 2 and 1/2 cookies for each

person] x people shared twenty cookies? [20/x cookies for each person]

What is a rule for finding the number of cookies each person will get if a people share b cookies? [b ÷ a cookies for each person]


Understand and use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems.


ReferencesCurcio, Frances R. and Nadine S. Bezuk. Understanding Rational Numbers and Proportions, 13-18, 24-27. Reston, VA: NCTM, 1992.

Happy Birthday to You

In this lesson, students look at patterns in calendars to determine the day of the week for a particular date. Students also use division to explain why those patterns occur, and then relate them to modular arithmetic. The final assessment is for students to find the day of their birth.

Learning Objectives

Students will:

Learn to perform modular arithmetic Use modular arithmetic to find the day of the week that they were born

Materials

America’s Birthday Activity Sheet America’s Birthday Answer Key Modular Arithmetic Activity Sheet Modular Arithmetic Answer Key

Launch

Ask students if they know their birthday. Students will chime in and share their birth dates with the class. Tell them that while that is the date on which they were born, you were asking for the day of the week on which they were born. If anyone knows the actual day they were born, have them share it with the class, and ask them how they can be sure. Tell them that they are going to develop a method for determining (or verifying, for those students who already know) the day on which they were born.

Explore

Pass out the America’s Birthday activity sheet to each pair. Have students work in pairs to find a pattern and answer the questions. For pairs that struggle, ask, “Does the day of the week change in any type of pattern?” [It moves forward one day most of the time. The other years it moved forward two days.] Encourage them to use the observations they make to draw a conclusion. [A particular date moves forward two days because of leap year.]

America’s Birthday Activity Sheet

Share

Call on pairs to share their conclusion. Have any pair with a different answer share their conclusion. Lead a discussion about why students may have seen different patterns in the calendar. Explain that students had to use their knowledge of leap years to help them draw the correct conclusion.

Give pairs five minutes to determine why the day of the week goes “forward” by one day every year and “forward” by two days every leap year. During this time, walk around and listen to students’ thinking. Identify a pair that is talking about the number of days in a regular year and the number of days in a leap

year. Ask, “How many weeks are there in a year? In a leap year?” [Most students will know that there are 52 weeks in a year.] Have students divide the number of days in a year by the number of days in a week to find the exact number of weeks in a year and leap year. [There are the same number of weeks with different remainders. A regular year has 365 days, which is 52 weeks, 1 day; a leap year has 366 days, which is 52 weeks, 2 days.] Have students relate the remainders they find to the Fourth of July pattern.

Tell students that mathematicians have a more sophisticated way of representing remainders with something called modular arithmetic. Using modular arithmetic, we say that 365 mod 7 = 1 and 366 mod 7 = 2. Ask, “What would 365 mod 52 and 366 mod 52 be?” [1, 2.] Ask them to explain why their answers are the same as 365 mod 7 and 366 mod 7. [Because 52 ×7 = 364, so 364 is a multiple of both 52 and 7.]

Pass out the Modular Arithmetic activity sheet to each student. Have a volunteer read the introduction to the class and summarize the explanation of mod. Have students complete the activity sheet individually. The back of the activity sheet will serve as their exit slip for the day. They will use the day of the week for their birthdays this year to determine the day of the week they were born. They must justify their answers.

Modular Arithmetic Activity Sheet

Summarize

Questions for Students

From year to year, what pattern helps to predict the day of the week for a particular date? [Dates move forward one day every year, except in leap years when they move forward two days.]

Why does a particular date not fall on the same day each year? That is, why does a date move forward one or two days from year to year? [The number of weeks in a year is not a whole number.]

How do the math problems 365 ÷ 7 and 366 ÷ 7 explain the calendar pattern for the day of the week on which a date falls from year to year? [The remainders tell how many days the date move forward.]

Explain what 365 mod 7 means. [The remainder when 365 is divided by 7.] What must you know to determine what day a date will be? [You must know the day the date is

for a specific year so you can work forward or backward.] Why might this method of finding the day of the week be cumbersome for dates a long time ago

or far in the future?

[It can be tedious to follow the pattern for a lot of years. However, once the pattern is known, you can identify the number of leap years and non-leap year between the current date and the date under consideration; multiply the number of leap years by two, and add the number of non-leap years; and then shift the the day by that many days from the same date in the current year. In addition, years that end in double zeroes (such as 1900, 1800, etc.) don’t follow a regular pattern for leap years. That may also cause problems for dates very far in the past or future.]

Assessment Options

1. Use the activity sheets for an assessment of student understanding.2. Have students find the day of the week for several dates, both in the past and in the future.3. Have student solve additional modular arithmetic problems.

Extensions 1. Have student find websites that tell the day of the week for any date entered. See if all the

websites agree for their birthday.

2. Have students research to find a formula for determining the day of the week for a particular date. They should create a mini-poster displaying the formula, explaining it, and then providing an example using their birthday.

The following formula is known as Zeller’s Rule:

f = k + [(13 × m – 1) ÷ 5] + D + [D ÷ 4] + [C ÷ 4] – 2 × C

where:

o [x] indicates the greatest integer contained within x. o k is the day of the month. As an example, if want to know the day for January 15, 2053,

then k = 15. o m is the month number, but months have to be counted specially for Zeller’s Rule:

March is 1, April is 2, and so on, so that February is 12. This makes the calculations simpler, but because of this rule, January and February are always counted as months 11 and 12 of the previous year. For our example, then, m = 11.

o D is the last two digits of the year. Because our example uses January (see previous bullet), D = 52, even though we are using a date from 2053.

o C is the first two digits of the year. In our case, C = 20.

Then, find the smallest positive integer d such that f = d mod 7. When doing this, use care if f is negative. For instance, if f = -13, then d = 1, because -13 = 1 mod 7.

Finally, a remainder of 0 corresponds to Sunday, 1 to Monday, and so forth.

For our example,

f = 15 + [(13 × 11 – 1) ÷ 5] + 52 + [52 ÷ 4] + [20 ÷ 4] – 2 × 20 = 15 + 28 + 52 +13 + 5 – 40 = 73

and 73 = 3 mod 7, so Jan 15, 2053, will be a Wednesday.

This formula could be entered into Excel as follows:

=A1+INT((13*A2-1)/5)+A3+INT(A3/4)+INT(A4/4)-2*A4

where A1, A2, A3, and A4 represent k, m, D, and C, respectively. The command INT() returns the greatest integer of the value inside the parentheses.

Of course, Excel has a built-in calculator to find the day for any date, so you could also enter a date in the form mm/dd/yyyy into A1, and then enter the following formula in a different cell:

=WEEKDAY(A1,2)

The returned value gives the day, with 1 = Monday, 2 = Tuesday, and so on.

Teacher Reflection Was your lesson developmentally appropriate? If not, what was inappropriate? What would you

do to change it?

How did your lesson address auditory, tactile and visual learning styles? How did the students demonstrate understanding of the materials presented? Were concepts presented too abstractly? too concretely? How would you change them? What were some of the ways that the students illustrated that they were actively engaged in the

learning process? Was students’ level of enthusiasm/involvement high or low? Explain why.

Big Math and Fries

Launch

We are lucky to live in an age where there is a lot of nutrition information available for the food we eat. The problem is that much of the data is expressed in percents and some of those percents can be misleading. This lesson is designed to enlighten students about how to calculate percent of calories from fat, carbohydrates, and protein. The calculations are made to determine if a person can follow the Zone Diet with only McDonald's food items.

Learning Objectives

Students will:

mathematically analyze the food they eat. identify the relationship between nutrients and the calories. perform calculations, including percents and conversions.

Materials

McDonald's nutrition facts (or any fast food chain) Computer with Internet connection (optional) Calculators Big Math and Fries Activity Sheet

Explore

In this lesson, students pick out a full day's worth of meals from a McDonald's menu with an eye towards the Zone Diet, which specifies percentages of fat, carbohydrates, and protein. The gist of the Zone Diet is that whenever you eat, you should strive to consume 40% carbohydrates, 30% protein and 30% fat. As a result, this diet has also been referred to as the 40-30-30 diet.

Many sources indicate that an average person requires about 2,000 calories per day. That number varies based on several factors, but is used as the target for this lesson. To prepare for the lesson, you might research some details on the Internet. For example, you can find out what various athletes consume in a day. You will find that it's much more than 2,000 calories. There was an urban legend floating around for a while that Michael Phelps (Olympic swimming gold medalist) was consuming 12,000 calories per day. That turned out to be untrue, but it might be interesting to start things off with a classroom discussion regarding how many calories various athletes consume. That could lead into a discussion regarding whether students know how many calories they consume and what proportion of nutrients are contained in the foods they eat.

Another lesson opener could be the portrayal of diet in the media. Some students may have seen the movie Supersize Me. Discuss the nutritional concerns about eating fast food. This can naturally progress into a discussion of diets. Suggest the idea of being "in the Zone." Ask whether any students have noticed a relationship between what they eat and how they feel. Do they sometimes feel sleepy? hyper? or just right?

Depending on time, you may wish to have a longer discussion about nutrition. If you choose to do this, you may wish to share the following information on nutrients with students:

Carbohydrates: our main source of energy Fats: one source of energy and important in relation to fat soluble vitamins Minerals: inorganic elements that are critical to normal body functions Proteins: essential to growth and repair of muscle and other body tissues Roughage: the fibrous indigestible portion of our diet essential to health of the digestive system Vitamins: important in many chemical processes in the body Water: essential to normal body function, both as a vehicle for carrying other nutrients and

because 60% of the human body is water

In nutrition, some information focuses on food weight and other information focuses on calories. Make students aware of this before they begin the activity to help them avoid errors based on these units. There is not a one-to-one relationship between food weight and calories. The Zone Diet percentages are all with regard to calories, so if you only have weight information, you need to convert to calories to match the Zone Diet percentages.

Fortunately, the conversions between food weight and calories are simple and students may have already studied this information in physical education or health class. Here are the conversions, which are also provided on the activity sheet:

Fat: 1 gram = 9 calories Carbohydrates: 1 gram = 4 calories Protein: 1 gram = 4 calories

This information can be written on the board or put up via a transparency.

Hand out the Big Math and Fries activity sheet and calculators. For this lesson, students should use calculators because of the number of calculations required. You can choose how many decimal places that they should round to. Just remember that some of their calculations will be converted from decimal to percent, so they'll need at least two decimal places for those calculations. Students should also be given McDonald's nutrition facts. You can either hand out paper copies or display McDonald's nutrition information via a computer projector.

Have students read the McDonald's nutrition pamphlet and try to pick out enough food so that the total number of calories adds up to 2,000 for the day. It is difficult to meet all the caloric and Zone diet requirements at once, so suggest to students to begin with only one or two. Students can attempt to ensure that their percent of calories from fat for the day is less than 30%. If students can do that, they've done well. Then challenge students who succeed to additional goals, such as keeping carbohydrates to 40% of the total calories and keeping protein to 30% of the total calories. It is difficult to achieve all three, but students should be able to keep fat under 30%. More advanced students may be able to get close to the proper percentage for all three nutrients. When more advanced students finish, have them help slower students who are not finished yet.

Share

A nice wrap-up for this lesson would be to have students that came closest to achieving Zone proportions present their findings and explain how they achieved their results. You might also have students could create posters to present their findings and explain why they would recommend the meal combinations that

they came up with.

SummarizeQuestions for Students

Were you able to stay under 30% for total calories from fat? Do you feel that you designed a healthy day of eating?

[Answers will vary]

What steps did you take in order to meet the requirements of 2,000 calories total and a 40-30-30 ratio?

[Answers will vary. What you're looking for are the strategies that students used to try to balance the results. For instance, did successful students focus on one nutrient, get the appropriate percentage and then change one food item to balance the other nutrients? Did they first calculate the total amount of grams needed for each nutrient base on a 2,000 calorie diet and then work backwards? Or did they come up with something new and unique?]

If you were not able to meet the Zone Diet requirements of 40-30-30, could you tweak a few items to change that? If so, which items would you change and how does that improve your carbohydrates-protein-fat ratio for the day?

[Students should look at nutrient percentages that are too high and try to figure out which items they could remove or replace in order to get better ratios.]

Would it be easier to design one Zone friendly meal and, if so, which items would you choose?

[Yes, it would probably be easier to design just one meal to meet the ratios. This should lead students to think about balancing nutrient ratios when they go to eat a meal or a snack.]

If you were to design the McDonald's nutrition pamphlet, what would you change from the current design?

[Answers will vary. One suggestion might be to provide percentage of each nutrient, not just fat.]

Assessment Options

1. Ask students to design a single meal and see how close they can get to the 40-30-30 ratio. 2. Allow students to design a day's meals using any food they choose to meet the Zone Diet.

Students should gather their own nutrition information and provide calculations for how they met the diet's restrictions.

3. Remove the caloric restriction from the activity. Just using the Zone Diet restrictions, is the activity easier, harder, or the same?

Extensions

1. Watch the movie, "Supersize Me," with the students and discuss how the movie relates to the lesson. Have students explain whether or not the movie was a fair representation and why. If your students are familiar with sampling and statistical analysis, you may also discuss the experimental model used in the movie.

2. Talk to a health or physical education teacher and see if they have a unit on nutrition where this lesson could be used as a complement.

3. This lesson is well suited for a spreadsheet application or a graphing calculator program. A spreadsheet application might take the grams of each nutrient for each desired item and automatically calculate the percentage of calories. Students could then more easily use trial and error to find the desired ratio of nutrients for a particular meal. This is only recommended in a class familiar with spreadsheets or graphing calculators.

Teacher Reflection

Were most students able to adjust their food choices to achieve Zone proportions? If not, what problems did they run into? How could these problems be avoided?

Were any students confused about the difference between calculating the calories due to the nutrient weights and calculating the percent of calories from a particular nutrient? If so, how could this confusion be avoided in the future?

Were students motivated to achieve Zone proportions or did they just pick various menu items to get the work part over with?

Did students feel that this lesson was interesting or of use to them?

NCTM Standards and Expectations

Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

1. Work flexibly with fractions, decimals, and percents to solve problems.2. Understand and use ratios and proportions to represent quantitative relationships.

This lesson was prepared by Michael Weingarden as part of the Illuminations Summer

Too Big or Too Small?

In this lesson, students develop number sense through a series of three hands-on activities. Students explore the following concepts: the magnitude of a million, fractions between 0 and 1, and the effect of decimal operations.


develop intuition about number relationships estimate computational results develop skills in using appropriate technology

Materials One thousand or more fake dollar bills (play money or rectangular sheets of

paper the approximate size of a dollar bill) Scissors One copy of Circle Template (on colored cardstock) for each student Calculators Decimal Maze Activity Sheet

Instructional Plan Included here is a selection of problems and activities, appropriate for the middle grades classroom, for which the underlying theme is the development of number sense. These activities can be used in varied ways to generate discussion and to extend student thinking about number-related concepts. The discussion that arises as students describe their thinking will certainly give insight into their thinking and will help in evaluating students' development of number sense.

Day1 Activity 1: Exploring The Size of a Million Dollars

This activity explores whether one million dollars will fit into a standard suitcase. If so, how large would the suitcase need to be? How heavy would it be? You may have students work in small groups (2 or 3 students per group) to explore these questions.

LaunchJust as you decide to go to bed one night, the phone rings and a friend offers you a chance to be a millionaire. He tells you he won $2 million in a contest. The money was sent to him in two suitcases, each containing $1 million in one-dollar bills. He will give you one suitcase of money if your mom or dad will drive him to the airport to pick it up. Could your friend be telling you the truth? Can he make you a millionaire?

ExploreInvolve students in formulating and exploring questions to investigate the truth of this claim. For example:

Can $1,000,000 in one-dollar bills fit in a standard-sized suitcase? If not, what is the smallest denomination of bills you could use to fit the money in a suitcase?

Share Have the student work in groups to figure out their answer and share their

finding with the class.

Calculators should be available to facilitate and expedite the computation for analysis.

Summarize1. What were some of the methods used to complete this activity?2. Have the student discuss what methods were more efficient than others.3. What formulas where used to figure out their results.

ExtensionsCould you lift the suitcase if it contained $1,000,000 in one-dollar bills? Estimate its weight.

Note: The dimensions of a one-dollar bill are approximately 6 inches by 2.5 inches. Twenty one-dollar bills weigh approximately 0.7 ounces.

Day 2Activity 2: Estimating Fractions Between 0 and 1

The model suggested here is easy to make and will help you evaluate your students' understanding of fractions between 0 and 1. Encourage students to make estimates using familiar benchmarks (e.g., ½, ¼, ¾).

MaterialsCircle Template

Copy the Circle Template onto light-colored cardstock.

LaunchGive each student a copy and ask them to cut out the circles and make a cut in the radius of each.

Have students put the circles together so that they can see the fractions printed on one side of one circle. Ask questions such as these:

ExploreShow a small part of the shaded circle (less than ¼). Can you name the part represented?Show a large part of the shaded circle (greater than ¾). Can you name the part represented?

ShareAsk students to reverse the circle with the printed fractions so that they cannot see the fractions. Ask students if they can:

Show a fraction that is a little bigger than ½. What name can you give it? Show a fraction that is between ½ and ¾. What name can you give it?

Continue asking questions that allow students to show their understanding of the fractions represented.

Other fraction models should also be used to evaluate students' understanding of fractions.

Have students create their own circles but now replace the fractions with decimals.

Summarize1. Can the student make correlations from fractions to decimals.2. Have the students share how they were able to make the conversion from

fractions to decimals.

3. Continue asking questions that allow students to show their understanding of the fractions to decimals.

4. Ask the students what were some things they learned from this activity.

Day 3 Activity 3: Exploring The Effect of Operations on Decimals

This activity provides an opportunity for students to explore the effect of addition, subtraction, multiplication, and division on decimal numbers.

LaunchWrite the problem (as described next) on the chalkboard or overhead. Ask students to discuss what they notice. Lead a discussion that focuses on these key points:

In computing the product of 4.5 and 1.2, a student carefully lined up the decimals and then multiplied, bringing the decimal point straight down and reporting a product of 54.0.

Reflection on the answer should have caused the student to realize the product was too big. Multiplying 4.5 by a number slightly greater than 1 produces an answer a little more than 4.5. Instead, this student applied an incorrect procedure (line up the decimals in the factors and bring the decimal point straight down) and did not reflect on whether the resulting answer was reasonable.

ExploreTell students that they will be playing a game to practice decimal operations and their effects. Encourage students to trace several paths through the maze while always looking for the path that will yield the greatest increase in the calculator's display. Note: Students often shy away form dividing by decimals less than 1, so you may want to discuss the general effect of dividing by a number less than 1 or multiplying by a number greater than 1.

Give each student a calculator and a copy of the Maze Playing Board activity sheet.Maze Activity Sheet Maze Playing Board Activity Sheet

Students are to choose a path through the maze. To begin, have the students enter 100 on their calculator. For each segment chosen on the maze, the students should key in the assigned operation and number. The goal is to choose a path that results in the largest value at the finish of the maze. Students may not retrace a path or move upward in the maze.

ShareIn pairs or in groups of three, students should discuss their strategies (after playing the game) and what worked best for them.

Students should be able to achieve a score in the thousands. The path highlighted below gives a result of roughly 6332.

Possible follow-up activities include finding the path that leads to the smallest finish number or finding a path that leads to a finish number as near the start number (100) as possible.

Summarize Understand the meaning and effects of arithmetic operations with fractions,

decimals, and integers. Develop and use strategies to estimate the results of rational-number

computations and judge the reasonableness of the results.

ExtensionsThe Decimal Maze can be modified depending on the level of your students and the topics covered in your classroom. For instance, the maze could be limited to positive whole numbers using only the operations of addition and subtraction for young students, whereas the maze could include scientific notation and exponents for older students. The Blank Maze can be modified to fit your needs

NCTM Standards and Expectations

Number & Operations 6-8


Develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results.

References Reys, Barbara J., et al. Developing Number Sense in the Middle Grades, 5, 8, 9, 22, 29, 41, 55, 56. Reston, VA: NCTM, 1991. Hoffer, Alan R., ed. Mathematics Resource Project: Number Sense and Arithmetic Skills. Palo Alto, CA: Creative Publications, 1978. Used with permission of the University of Oregon. Morris, Janet. How to Develop Problem Solving Using a Calculator. Reston, VA: National Council of Teachers of Mathematics, 1981.

Fraction Circles NAME ___________________________ To create a fraction circle: Cut out both circles

Make a cut in each circle along the radius that says Cut. Insert one circle into the other. Rotate the circles around the center to show fractions of different sizes.

© 2008 National Council of Teachers of Mathematics http://illuminations.nctm.org

The Cost of Being LateWe are bombarded in the media with ads offering 0% interest or teaser rates of 2.9%. These ads are devised to entice us to sign up for these limited time offers that the companies tell us would be crazy to miss. The goal of these ads is to get us to use credit to buy on impulse. If we take the time to analyze the offer, we might realize that if it sounds too good to be true, then it probably is. In this lesson, students will work through a credit card scenario with a teaser rate, minimum payments, fees, and rate increases for being late.

Learning ObjectivesStudents will:

Experience the impact to the overall cost of borrowing on credit when Interest rates increase Fees are assessed for late payments Minimum payments are made Modify the scenario to optimize payments for quicker repayment Modify the scenario to determine the impact of multiple late payments

Materials Copies of print ads promoting low or no interest on the purchase of goods or the

use of credit such as: 0% APR for 72 months 2.9% APR for 60 months Zero down and 0% interest for 12 months Computer with spreadsheet application Chart paper and markers (optional for presentations) Cost of Being Late Spreadsheet Cost of Being Late Activity Sheet: Spreadsheet Application version

LaunchGetting your first credit card can be very exciting, but unfortunately some people do not take the time to consider the costs associated with buying on credit. A credit card can make it easier to buy on impulse, and we do not always take the time to consider the hidden costs of buying on credit. This lesson will help kids understand the perils of making impulse purchases on credit, paying only the minimum each month, and/or making late payments.

In this lesson, students will work individually or with a partner to answer a series of questions related to buying on credit. They will first work under the assumption that they are not paying off the balance but will make the minimum payment and revolve the debt. Once they have seem how much the cost of borrowing on credit and making minimum payments effects their overall cost of the items purchased on credit, they will rework the scenario to formulate an optimal payment plan for paying off the debt within a time from of six to twelve months.

You may want to walk through some practice scenarios to ensure your students have a working understanding of how interest is calculated.

The primary goal of this lesson is to understand that the costs associated with buying on credit and that making only minimum payments is problematic to long-term financial health. The secondary goal is to formulate a plan to extinguish the debt quicker. This lesson will help them see the real cost of purchasing an item when interest and fees are added to the purchase price. The scenario that will be used for this lesson is based on the first year of the life of a new credit card. The student handout describes the scenario and posses a series of questions based on the use of the spreadsheet application.

ExploreLesson Opener In order to assess the level of consumer finance knowledge your students posses, consider asking students to discuss the following questions in small groups or as a whole class:

What does it mean to purchase something on credit? o [Answers will vary, but what you are looking for is to see if students know

that interest and/or fees can be associated with purchasing on credit. Help them understand that unless a credit purchase is paid in full on or before the due date that the credit granter will charge them interest (rent for the use of the credit grantors money).]

o Show students some ads that depict unrealistic interest rates or zero interest for extended periods of time. Car ads are a great visual tool since they often times advertise unrealistic rates over long periods of time.

How can a merchant stay in business if they are lending money to consumers to purchase goods but not charging interest for the use of that money over time?

o [Answers will vary from students not having any idea to other students surmising that the price of the merchandise is higher to reflect the cost of borrowing. This is correct and often times in the fine print of the add there will be a disclosure indicating a discount for cash purchases.]

What is the difference between simple interest and compound interest? Which do credit card issuers use?

o [Simple interest is the amount of interest based only on the principal amount owed. Compound interest is interest calculated on the sum of the principal and any previous interest. Credit card issuers typically use compound interest. How exactly interest is calculated will vary with credit card issuers. For more detail on how interest is calculated, see the disclosure statements provided by various issuers. Note: The reading level of most credit disclosure statements is beyond the comprehension of most middle school students.]

What is meant by minimum payment and who determines the minimum? o [Some students will be under the impression that they can determine how

much they want to pay. This is not necessarily true. The minimum payment is usually a percentage of the outstanding balance or a flat rate

that must be paid on or before a specified date. In 2005, the minimum payment was changed from 2% to 4%]

ShareAs a whole class, ask students to share what they knew about credit cards. Expect that they may share family experience or repeat what they have heard in the media. If it does not come out in the class discussion, make sure students understand that the use of a credit card creates debt and a legal obligation to pay while a debit card takes money from an existing checking or savings account to pay for purchases and does not result in a debt obligation. The terms debit card and credit card are often times mixed up by students and even some adults. If you have not already done so, this would also be a good time to share some the ads offering teaser rates or extended payment options. Be sure to point out the fine print that is usually at the bottom of these ads. The fine print will often disclose the true cost of credit based on a simple scenario.

The scenario we will use for this activity includes the charging of three retail purchases and a dinner for friends. Distribute either the Cost of Being Late: Compound Interest Simulator or Cost of Being Late: Spreadsheet Application activity sheet, depending on which method students will use.

For simplicity, no additional charges will be made over the next twelve months. The scenario will be based on a teaser rate of 10% for six months and a default rate of 22% thereafter. (The default rate is the published rate without benefit of any promotional offers. It is sometimes referred to as the posted rate.) One payment will be made late resulting in a late fee of $40 plus an increase in the APR from 22% to 28%. Our goal will be to determine the balance due at the end of one year along with the cost of using credit for that twelve-month period. If your students are struggling with the concept of compound interest you may want to consider having them work with a partner pairing a weaker student with a stronger student.

Spreadsheet Application

Use the Cost of Being Late spreadsheet. The application provided with this activity has the formulas built in and is based on a twelve month scenario using the mp3 example: Have students consider the purchase of an mp3 player at a cost of $200 with payments of $20 per month and an interest rate of 22% APR. Have students enter the information into the simulator as follows (Note: Only whole numbers can be entered into the dollar fields, which means kids will need to round up or down using standard rounding rules):

It will take twelve months to pay for the player for a true cost of $222.98. That true cost will be even higher if payments are not made on time and late charges are assessed or if interest rates are increased before the debt is paid in full. In this example the cost of buying the player on credit was $22.98. Consider asking students what they could have done with that $22.98 if they had not bought the player on credit and instead waited to pay cash.

A minimum payment of 4% or $20, which ever is greater is standard but since it is an open field you can have the kids work with a variety of scenarios. You might consider having them see how long it will take to pay off if they make minimum payments of $10 or if the rate were higher or lower. Depending on the skill level of your students and time constraints you might ask the students to pose an additional example that they could use the calculator for such as a new game system or some other consumer purchase that may be of interest to them. Some students may want to simulate the purchase of a car which can be done but I would recommend using the purchase of a big ticket item such as a car as an extension activity.

Retail Credit Card Purchase of an mp3 Player Example Payment

# Initial

Balance Purchases Fees Monthly Payment

Annual Interest Rate

Interest Charges

– $200.00 – 22% $3.671 $203.67 – – $20.00 22% $3.372 $187.03 – – $20.00 22% $3.063 $170.10 – – $20.00 22% $2.754 $152.85 – – $20.00 22% $2.445 $135.28 – – $20.00 22% $2.116 $117.40 – – $20.00 22% $1.797 $99.18 – – $20.00 22% $1.458 $80.63 – – $20.00 22% $1.119 $61.75 – – $20.00 22% $0.7710 $42.51 – – $20.00 22% $0.4111 $22.92 – – $20.00 22% $0.0512 $2.98 – – $2.98 22% $0.00

Totals – $222.98 $22.98

Answers Simulator Spreadsheet 1. $532.00 $532.002. $516.43 $516.433. $436.63 $436.634. $419.43 $419.115. $128.21 $127.11

http://illuminations.nctm.org/Lessons/CostofBeingLate/CostofBeingLate-SS.xls

SummarizeQuestions for Students

1. If the interest rate were cut in half, would the time to pay off the debit also be cut in half?

a. [Expect some students answer yes to this question. If they do, I would recommend projecting the simulator and entering a rate that of 11% for the mp3 player example. At 22% the balance due in the 10th month will be $22.51 while at 11% it will be $10.65. The overall cost of credit drops from $22.98 to $10.75, which is a change of $12.23 or 53%. You may want to ask students why the interest paid was cut by more than half when the interest rate was cut by exactly half. The answer is the interest was reduced by more than half because we did not change the minimum payment amount of $20 therefore more was applied to the balance with the lower rate.]

2. If we doubled our payment from $20 per month in the scenario to $40 per month, how long will it take to pay off the debit?

a. [If we leave in the late payment fee and increased interest rate then will will take approximately 16 months to pay the debt in full.]

3. Does interest added to your balance reduce your available credit line?a. [Yes, your outstanding balance includes principal balance, interest, and

fees. Your available credit is your credit limit less this balance. It is possible to incur an over limit charge if the applied interest puts you over your credit limit.]

4. Can you make more than the minimum payment?a. [Yes, your payments must be equal to or greater than the minimum

payment. You can pay off your balance ahead of time as well as make payments before the due date.]

5. Can you re-negotiate your interest rate?a. [Yes, it is sometimes possible but these negotiations are handled by the

lender on a case-by-case basis after taking into consideration the borrowers credit history.]

6. Why is it better to have a lower rate of interest?a. [Answers may vary but what you are looking for is for students to

recognize that if the rate is lower that more of their minimum payment will be applied to the outstanding balance. It also means that the cost of buying on credit is less. Buying on credit is like paying rent to use someone else’s money. The lower the rate (rent) the lower the cost is to borrow.]

Assessment Options Collect the activity sheets and check for completeness. Have students do presentations of their overall findings. Modify some of the parameters of the activity, such as initial spending and missed

payments, and have students redo the activity.

Ask students to simulate the worst case scenario where no payments are made and calculate how much is owed after a year.

Have students collect data on their own spending and pretend they paid it all with a credit card, then calculate the cost as they did in the activity assuming they made only the minimum payment every month.

Extensions1. Invite an issuer of credit cards to come in and talk about the process of apply for

and gaining approval for a credit card.2. Have students collect disclosure statements from various credit card issuers to

determine how they calculate the balance from which interest will be calculated. This extension is only recommended for teachers who are comfortable with consumer finance and disclosure statements as it can raise many questions that could prove difficult to answer. Also, the reading level of many disclosure statements is beyond that of the typical middle school student.

3. Invite a consumer credit counselor to come in and share stories of people’s lives that were changed for the worse due to misuse of credit cards. This could be a panel consisting of a consumer credit counselor, bankruptcy attorney, and debt collectors.

4. Alter the scenario to include additional purchases made in the first year, cash advances, and/or additional charges for late payments and being over the limit.

5. Have students interview an adult who uses credit cards to determine what advice they would have for a person who had just received their first card.

6. Have students’ research how a credit card issuer calculates the interest due each month and what amount is used as the basis for the calculation.

In its simplest form, credit card issuers take the outstanding principal balance plus any new charges and/or fees, less any payments or credits as of the statement cycle date then they calculate the interest due using the formula Interest = principal×rate×time. For example:

Outstanding principal balance = $340New charges = + $75Fees or pervious months interest = + $10Payments = – $25Balance on cycle date = $400Assume an annual percentage rate (APR of 12% or 0.12)Assume time of one month (1/12)I = ($400)(0.12)(1/12) ≈ $4The outstanding balance due is now $404.

By repeating this process each month we are compounding interest. Interest rates are always quoted as annual percentage rate (APR). To determine the rate charged for one month you would need to divide the annual percentage rate by twelve. Some credit card issuers will use average balance or blending of average and actual balances to determine the basis for calculating interest. Methods can vary greatly between credit card issuers. This lesson has been simplified for the age

level of the students. For more information on actual practices by credit card issuers see the Federal Reserve Board

Teacher Reflection Did students have the background knowledge necessary to complete this activity?

If not, what changes would you make for the next time this activity is used? Was students’ level of enthusiasm/involvement high or low? Explain why. Did you challenge the achievers? How? How did the students demonstrate understanding of the materials presented? What were some of the ways that the students illustrated that they were actively

engaged in the learning process? Did you find it necessary to make adjustments while teaching the lesson? If so,

what adjustments, and were these adjustments effective? What worked with classroom behavior management? What didn't work? How

would you change what didn’t work?


1. Work flexibly with fractions, decimals, and percents to solve problems.

This lesson was prepared by Julie Healy as part of the Illuminations Summer Institute.

7th grade - grubbs-math-masterminds.wikispaces.com€¦ · web viewduring this unit student will...

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