November 20135th and 6th Grade

Please make a name tent

Group CommonalitiesOn a blank sheet of paper, draw a vertical and horizontal line, cutting the paper into quarters.

Number each one of the sections 1 - 41

2

3 4

Group Commonalities•State something that you like

•Find out how many in your group also like it

•However many people in your group like it, write what it is in that numbered spot

•Pass the paper to the next person and continue the steps above for about 5 minutes

How Many Noses Are in Your Arm?

•The Statue of Liberty’s nose measures 4 feet 6 inches from the bridge to the tip. •How long is her right arm?•Use tape measures, calculators and what you know about body parts to determine the length of the Statue of Liberty’s right arm, the one holding the torch. •Decide on a problem solving strategies and to record it. •How long is her arm?(The actual length

of her arm is 42 feet.)

Share Solution Strategies

•Look through the handout of questioning students ▫What questions could you (have) asked

Before During After

Ratios, Proportions, and Percents5th and 6th GradeNovember 2013

Common Core Standards

•On a piece of paper:▫Write one thing going well▫Write one thing you are concerned about

Mathematical Practice

•Including in your lessons?•Are the placemats working?•How are you implementing them?

Standards

•Look through district and state standards being addressed today

Work Through…

•Pattern Block handout AND•Lesson 2 handout•Compare the two handouts in terms of

questioning format ▫What are strengths and weaknesses of each

handout?

Proportion Card Game• Remove all face cards• For the purposes of this game, aces = 1• One player shuffles the deck of cards and deals out 4

cards to each team. The remaining cards should be placed face down in the center of the table.

• Teams will take turns drawing a card from the deck and discarding one from their hand.

• Each time a team finds a proportion in their hand, they should call out “Prop!” Then, they lay down their cards on the handout in the proper places to make a true proportion statement.

• For each correct answer, teams earn a point.• Each time a team wins a point, both teams should

discard their cards, the deck should be shuffled and both teams should be dealt new hands of 4 cards.

I. Introduction to Ratio and Proportion

What is a ratio?    What are some examples of ratios?  What is a proportion?  

•The quantitative relation between two amounts showing the number of times one value contains or is contained within the other•A statement of how two numbers compare

# of girls to # of boys in this room# of teachers in Roosevelt to # of teachers in Arizona# of tables in this room to # of chairs in this room

A statement about 2 equal ratios

Analogy:   Proportion:   The proportion above can also be written:

A nest is to a bird as a den is to a fox.

2 is to 4 as 6 is to 12.

2:4 as 6:12 2:4 = 6:12 2 6

4 12

=

According to Piaget’s theory of intellectual development proportional reasoning is an important skill for children to develop. Proportional reasoning is something that is built through an individuals’ experience and takes time to develop. The following diagram illustrates the concepts which are an important components of proportional reasoning.

To assist students’ development of proportional reasoning, it is not sufficient to teach a section on each of these topics to develop proportional reasoning. 

Relative ThinkingRelative thinking can be described as

thinking multiplicatively. It is important for students to understand both absolute change and relative change. Making the transition from absolute change to relative change is an important step in transitioning from additive to multiplicative reasoning. x 1 2 3 4y 6 12 18 24

y = 6x

Partitioning

The partitioning of an object is the process of dividing the object into a number of disjoint parts that collectively make the whole. When dealing with fractions, to determine fractional parts, one partitions the object into parts of equal size.

UnitizingUnitizing is the cognitive process used to assign

a unit to a given quantity. For example, when asked to think about a case of soda, do you picture 24 cans, two 12- packs, or four 6-packs [Lamon, 2006]? It is desirable for students to build flexibility in determining the size of the chunk of a quantity that they use for a unit, as different situations may call for different sized chunks. Asking students to explain their choices can help encourage this flexibility.

UNIT RATEIf 3 cost $5 how much will 1 cost?3 ÷ 5 = 0.6

Ratio Sense

Ratio Sense involves the ability to think flexibly in problem situations involving ratios.

Rational Numbers

Rational Numbers are numbers of the form a/b , where b ≠ 0 and a and b are b integers. That is, numbers that can be written as fractions. More importantly, reasoning with rational numbers requires the ability to reason flexibly with fractions, ratios, rates, and percents, and the operations on them.

Quantities and Change

Quantitative reasoning involves the ability to interpret and operate with change. This may require operating with constant or varying rates of change.

y = 6x

6 is the constant

The ability of students (and people, in general) to distinguish between relationships which are proportional and relationships which are non-proportional is a key aspect of proportional reasoning.

For this reason, students should be exposed to situations which are proportional and non-proportional to learn when it is appropriate to use a multiplicative solution strategy.

These types of experience are important because in the absence of such experiences students tend to inappropriately model non-proportional problem situations with proportions.

One of the below questions involves proportional reasoning the other does not, which does and which does not? Discuss in your groups

 Question 1: Kris and Rich like to skate laps together

around an ice rink since they both skate at the same constant rate. Today, Kris started skating first. By the time that Kris had completed 9 laps Rich had completed 3 laps. How many laps will Kris complete by the time that Rich completes 15 laps? Explain.

   Question 2: If 3 bags of mulch weigh 21 pounds, how

many pounds will 8 bags of mulch weigh? 

The temptation in Question 1 is to set up a proportion to solve the problem, since there are three known quantities and one unknown quantity, resulting in 45 for the answer as shown below.

  9 x 9 x 15 = 3 x x 3 15 45 = x

 The thinking behind this approach is that 9 laps for Kris is to 3 laps for Rich as x laps for Kris is to 15 laps for Rich. However, this particular problem can not be represented by setting up a proportion, so this approach is incorrect.

To see that the above approach will not work, reflect on whether or not 45 laps is a reasonable answer.

=

Let‘s take a closer look at this analysis of the situation by making a table to represent the information in the problem.

# of laps Rich completes

# of laps Kris completes

3 9

4 10

5 11

6 12

7 13

8 14

9 15

10 16

11 17

12 18

13 19

14 20

15 21

The correct answer, 21 laps, can be seen by making a table.

In Question 2, it is implied that we are dealing with the same size bags of mulch. Since 3 bags of mulch weigh 21 pounds, 1 bag of mulch weighs 7 pounds. Hence, when determining the weight of 8 bags of mulch ones needs to only multiply the number of pounds per bag by the number of bags (i.e., 7 lb/bag × 8 bags = 56 lb). Thus, the situation can be modeled by the linear function y = 7x, where y represents the total weight of x bags.

y 21 y x 3 = 21 x 88 3 3y = 168

y = 56

=

Can proportions be used to solve either of these problems? Why or why not? Discuss in your groups

 Question 3: Alisa is painting her living room.

She can paint the entire living room in 4 hours. Assuming that Karen can complete the job in the same amount of time as Alisa, how long will it take Karen and Alisa to paint the living room together?

   Question 4: If one player on a soccer team

weighs 170 pounds, then what is the weight of 4 players?

Question 3: Alisa is painting her living room. She can paint the entire living room in 4 hours. Assuming that Karen can complete the job in the same amount of time as Alisa, how long will it take Karen and Alisa to paint the living room together?

Common thing to do is set up the proportion:

1 person 2 people 4 hours ? hours

Which would give an answer of 8 hours. Is that reasonable?

Can not use a proportion to solve this problem

=

Stopping to evaluate whether or not 8 hours is a reasonable answer quickly leads one to conclude that this problem can not be solved by setting up a proportion. To see the correct answer, one notices that both people can complete the job alone in 4 hours so it should take half as much time for the two people to complete the job together, or 2 hours. Another way to analyze the problem is to use a unit rate strategy – each person can complete 1 job every 4 hours so let‘s assume that each person can complete ¼ of a job per hour.

Understanding Ratios and Proportions A. Consider the following situations and: Determine whether the problem being

modeled is a proportion. Explain your reasoning. If it is a proportion, solve the problem  1. Ray and Crystal invested money in a

business and will split the profit in a ratio of 2:3. If the profit from the business is $1000, how much money will each person receive?

2. Can Courtney enlarge a 3 inch by 5 inch photograph proportionally to a 4 inch by 6 inch photograph? Is this possible?

 

3. A fourth-grade class needs 5 leaves each day to feed its 2 caterpillars. How many leaves will they need each day to feed 12 caterpillars?

   4. If it takes 6 construction workers 4 days to complete a

job, how long will it take 2 construction workers to complete the job? [Assume that each construction worker can complete the entire job in the same amount of time when working alone—they work at the same rate.]

 

1. 2 Ray 3 Crystal Ray: $400 5 1000 5 1000 Crystal: $600

2. No, a proportion can not be used as they are not equal ratios 3 4

5 6

3. 2 12 2x = 5 (12) x = 30 leaves 5 x

4. workers 6 2 x = 1 1/3 – Not Reasonable

days 4 x A proportion cannot be used

= =

≠

=

=

Jeopardy

Percents

•Work through both sides of the handout with your table

•What are the objectives being met with these questions?

SECTION 1: THE MEANING OF PERCENT

In your own words, what is the meaning of percent?

Realistic or Unrealistic?1. George ate 130% of his breakfast.2. Callie ran 2 miles per hour yesterday. Today she ran 50% faster.3. Ming found a coat on sale. It was marked 100% off.4. Matthew got a pay raise of 5%.5. Nancy plays the state lottery. She figures she either wins or loses, so she has a 50% chance of winning?6. Mary went on a diet. She lost 80% of her previous weight.7. Tom’s boss gave him a pay cut of 50%. Then the boss gave him a raise of 50%. Tom figures he now is making the same salary.8. Carolyn baked a cake. She gave 40% to her sister, 30% to her mother, and kept the other 30% for herself.

Unrealistic

Realistic

Realistic

Unrealistic

Unrealistic

Unrealistic

Realistic

What percent is the exact equivalent of 1/3?

Percent Can Make Sense

• This activity is designed to help students develop an understanding of the relationships between fractions and percent.

• The use of a particular visual representation that is explicitly related to the meaning of per cent helps not only to reinforce a conceptually oriented understanding of percent but also suggests a mental math method for the conversion of fractions to percent. Now to examine a model of percent and its connections to reasoning

What does ¼ mean?

1 out of every 4. Use one of the hundreds tables to color one out of every group of four squares

No Calculators!

•Use the other hundreds tables to represent the following fractions:

1/5, 1/2, 3/4, 1/8

1/8 = ?

These 4 are remaining. What do we do with them?

12 and 4 left over?

12 4/8 = 12.5%

Still No Calculators

Now try:

1/9, 1/12, 1/3

1/9

11 squares shaded, 1 left over out of the 9.

11 1/9 %

Examples

•Change 125% to a fraction.

125% means , which simplifies to

•Change 0.005 to a percent.

125

100

5

4

5 0 50 005 or 0 5

1 000 100

.. , . %.

,= =

Convert to PercentsWrite each of the following as a percent.

1. 0.03 2. 0.33% 33.3%

3. 1.2 4.0.00042120% 0.042%

5. 1 6.100% 60%

7. 8.66.6%

Convert to DecimalsWrite each of the following percents as a decimal.

1. 5% 2. 6.3%0.05 0.063

3.100%

4. 250%1 2.5

5. 6.0.006 0.3

SECTION 3: CONVERTING BETWEEN FRACTIONS-, PERCENTS, AND DECIMALS—STUDENT RESOURCES

http://www.khanacademy.org/search?page_search_query=fractions+decimals+percents

Ratio and Proportion Questions

•Questions come from Questioning book discussed last session

Georgia lessons

•Look through the lessons/activities from the state of Georgia

Feedback

•Please complete the ½ sheet