2 Day Training Day 2

Day One Slide 2

Checking In Rates, Part 1 Explore Chapter 4

Using ReportsIn the Lab Next StepsReview

Evaluation

Lesson 17 Lunch PacingRates, Part 2

Diagnostic Assessment

Follow the map to student success.

Challenge

SuccessSuccess

Success

Day One Slide 4

Checking In Rates, Part 1 Explore Chapter 4 Break

EvaluationDiagnostic Assessment

Lay the foundation for a powerful model with many applications.

One Day Training Slide 6

Content Experience: Rates

Rates.exe

Day One Slide 7

Exploring Lessons 15 & 16

Use chapter resources to find the big ideas and key concepts.

Chapter FourPage 143 C

Use measures expressed as rates and measures expressed as products to solve problems.

7MG 1.3

Convert one unit of measurement to another.

6AF 2.1

Converting Units

Chapter 4

Lesson 15

• Use measures expressed as rates.

• Use rates to describe proportional relationships.

• Perform unit conversions using rates.

Objectives

Remember from Before

• How many meters are in a kilometer?

• How are units used to measure quantities?

1. How many kilometers is 10,000 meters?

2. What number is 4 times the value of 12?

3. Use mental math to solve these equations.

a. 4 × a = 12

b. 5 × b = 35

c. 14 + 25 = c

d. 1,020 + d = 1,725

Get Your Brain in Gear

The following situation uses units of feet and inches:

We can represent “1 foot” on the number line like this:

We can represent “3 inches” on a different number line like this:

But how do we represent both “1 foot” and “3 inches” on the same number line?

Unit Conversion

My dog is 1 foot and 3 inches tall.

Since 1 foot is equal to 12 inches, we can convert the above situation to:

My dog is and 3 inches tall.

Now that both values use the same units, we can represent the dog’s total height on a single number line:

My dog is and 3 inches tall. 1 foot

12 inches

1. Which number line shows the equivalent of “1 yard and 2 feet”? (Remember that 1 yard converts to 3 feet.)

Check for Understanding

We write the conversion rate from feet to inches like this:

We read this rate as “12 inches per foot”.We write “1 foot” below the line because we are converting FROM feet.We write “12 inches” above the line because we are converting TO inches.

We always write what we are converting FROM on the bottom and what we are converting TO on the top:

2. For each rate identify what we are converting FROM and what we are converting TO:

a. 1 dollar10 dimes

36 inches3 feet

d. 24 hoursc. 1 day

7 days1 week

Check for Understanding3. There are 3 teaspoons in a tablespoon. Write the rate

from teaspoons to tablespoons.

4. There are 1,000 meters in a kilometer. Write the rate from meters to kilometers.

Here is a rate from “cars” to “wheels” (assuming 1 car has 4 wheels):

Here we have 3 cars:

How many wheels does this correspond to?

After the conversion, we can count the wheels to find that 3 cars corresponds to 12 wheels:

To go from wheels back to cars again, we invert the rate:

“4 wheels” is below the line because we are going FROM wheels.“1 car” is above the line because we are going TO cars.This rate means that each group of 4 wheels gets replaced by 1 car.

We get our 3 cars back.

FROM1 car

4 wheels

5. Each spider has 8 legs.

a. Write the rate from legs to spiders.

b. Write the rate from spiders to legs.

6. Five dimes converts to 2 quarters.

a. Write the rate from dimes to quarters.

b. Write the rate from quarters to dimes.

7. Ten pennies converts to 2 nickels.

a. Write the rate from pennies to nickels.

b. Write the rate from nickels to pennies.

My grandma has 3 grandchildren. A couple of winters ago she knitted all of us socks. She used the following rate to go from units of “grandchildren” to units of “socks”:

This rate means that 1 grandchild corresponds to 2 socks. Let’s perform the conversion on the 3 grandchildren:

Here’s another example.

Let’s look at another example.

If it takes 3 legs to make a tripod, how many tripods can we build out of 15 legs?

We need the rate that relates legs to tripods.

Now let’s use this rate to turn the 15 legs into tripods:

Since we are going TO tripods, we place “1 tripod” above the line.

Since we want to go FROM legs, we place “3 legs” below the line.

We see that we can make 5 tripods out of 15 legs.

8. A pentagon has 5 sides. Which rate properly goes FROM pentagons TO sides? Which rate properly goes the other direction, FROM sides TO pentagons?

Consider the following situation:

The store near me sells 2 pineapples for 3 dollars. How many pineapples can I buy for 9 dollars?

To answer this, we need to go FROM dollars TO pineapples. Here is the rate:

Let’s use this rate to turn 9 dollars into pineapples:

I can buy 6 pineapples for 9 dollars.

2 pineapples

3 dollars

9. If 7 bananas cost 2 dollars, what is the rate from dollars to bananas?

How many bananas can you buy with 6 dollars?

How many can you buy with 8 dollars?

10. If 4 peaches cost 3 dollars, what is the rate from dollars to peaches?

If I spent 6 dollars on peaches, how many did I get?

How many peaches would I get for 9 dollars?

Consider the following rate:

This rate turns 5 squares into 2 triangles.

Let’s use this rate to turn 20 squares into triangles:

Using the above rate, 20 squares become 8 triangles.

11. Turn 16 squares into triangles using the following rate:

12. Turn 6 circles into stars using the following rate:

Multiple Choice Practice1. There are 16 ounces in a pound. Which rate

converts from ounces to pounds?

1. Write the rate FROM 16 tablespoons TO 1 cup.

A student made 2 mistakes below. Find and correct each mistake.

Find the Errors

2. Convert the following peanuts to pennies using this rate:

Finding Rates from Situations

Chapter 4

Lesson 16

Use measures expressed as rates and measures expressed as products to solve problems.

7MG 1.3

• Use rates to describe proportional relationships.

• Describe rates using words and symbols.

• Use rates to solve problems.

Objectives

• What is a rate?

• How are rates written in symbols?

d. m = 5

a. 2 × a = 18 b. 3 × b = 18

c. 49 = c × c d. 30 = 6 × d

2. Find the value of m + m + m + m when:

a. m = 2 b. m = 3

c. m = 4

It costs 3 dollars to buy 2 loaves of bread. How many loaves can you buy with 9 dollars?

and “2 loaves of bread” on the top:The rate is going to have “3 dollars” on the bottom

TOFROM

In this lesson, we will learn how to figure out the rate from the description of a situation.

Since we want to find “how many loaves”, the problem is asking us to go FROM dollars TO loaves of bread.

It costs 3 dollars to buy 2 loaves of bread. How many loaves can you buy with 9 dollars?

Let’s perform the conversion to answer this question:

We can buy 6 loaves of bread for 9 dollars.

TOFROM

Since we want to find the cost, we need to go FROM loaves of bread TO dollars. We need to invert the rate.

Use the same price of 3 dollars to buy 2 loaves of bread. How much does it cost to buy 8 loaves of bread?

Let’s try a slightly different situation:

Now let’s perform the conversion:

It costs 12 dollars to buy 8 loaves of bread.

1. Find the rate then perform the conversion:

a. The price of 7 apples is 2 dollars. How many apples can you buy with 6 dollars?

b. It costs 5 dollars to buy 4 avocados. How much does it cost to buy 8 avocados?

c. It costs 1 dollar to buy 1 bag of chips. How much does it cost to buy 459 bags of chips?

d. It costs 20 dollars to buy 3 coconuts. How many coconuts can you buy with 40 dollars?

When we ask someone to give us 4 quarters in exchange for a dollar, we are converting from dollars to quarters.

2. Find the conversion rates to make the following exchanges:

a. from nickels to dimes

b. from pennies to nickels

c. from quarters to nickels

d. from dimes to dollars

e. from dimes to nickels

f. from dollars to dimes

There are many ways we use language to talk about rates. Consider the following rate:

Here are just some of the ways we might communicate the above rate using language:

• 3 meals per day• 3 meals every day• 3 meals each day• 3 meals in a day• 3 meals a day

All of these refer to the same rate.

Sometimes a rate is stated in one way, but actually the inverted rate is required.

Honey bees fly 7 miles per hour. At this rate, how long would it take a bee to fly 21 miles?

The situation states a rate of “7 miles per hour”, which is written as:

But the question asks us to go FROM 21 miles TO hours which means we actually want the inverted rate:

3. Find the rate from the situation:

a. There are four seasons in a year. If 20 seasons went by, how many years is this?

b. There is one summer Olympics every 4 years. How many summer Olympics in 20 years?

c. The shelter gave out 5 blankets for every 3 people. They gave out 30 blankets, how many people got them?

d. Each box can hold 20 glasses. There are 200 glasses. How many boxes are needed?

Here is a fish and a shoe:

What is the rate from fish-lengths to shoe-lengths?

Because the fish is longer, we need fewer fish-lengths to equal the same distance measured in shoe-lengths.

To find out, measure the same distance using fish-lengths and shoe-lengths.

3 shoe-lengths is equal to 2 fish-lengths.

We can create the following rate:

Let’s use this rate to answer the following question:

We find that 4 fish-lengths is the same as 6 shoe-lengths.

An object is 4 fish-lengths long, how many shoe-lengths long is it?

To answer this, we simply use our rate to convert from fish-lengths to shoe-lengths:

4. What is the rate from car-lengths to truck-lengths as shown here:

Using this rate, convert 15 car-lengths to truck-lengths.

My friend’s company has 8 employees who ride their bikes to work.

At my friend’s company, 2 out of 3 employees ride their bikes to work. There are 12 employees at the company. How many bike to work?

What is our rate?

Let’s perform the conversion:

5. Two out of 5 of the books on the shelf are over 600 pages long. If there are 15 books on the shelf, how many have more than 600 pages? How many don’t have more than 600 pages?

6. In the design, 1 out of 3 shapes were rectangles. There were 6 rectangles. How many shapes were in the design?

7. The manager only worked 4 out of 7 days last February. If there were 28 days in the month, how many days did the manager work last February?

Check for UnderstandingFind the rate then perform the conversion.

Let’s use this rate to perform the conversion on the 6 donuts:From this we see that Homer ate 2 bananas.

Homer ate 3 times as many donuts as bananas. If Homer ate 6 donuts, how many bananas did he eat?

“3 times as many donuts as bananas” means there are 3 donuts for every 1 banana. How many bananas tells us to go from donuts to bananas so we use the following rate:

8. There were 4 times as many dogs as people at the park. If there were 12 dogs, how many people were there?

9. The student got 5 times as many answers right as wrong. If the student got 7 wrong, how many did the student get right?

10. At the party there were twice as many students as teachers. If there were 10 students, how many teachers were at the party?

Find the rate then perform the conversion.

Multiple Choice Practice1. If you can get 8 dollars an hour, how long does it

take to earn 32 dollars?

A student made 1 mistake answering the questions. Find and correct the mistake.

1. The bug walked 3 meters every 5 minutes. How long did it take to walk 6 meters?

2. He was sick 4 out of 5 days during the break. If the break was 15 days, how many days was he sick?

Find the Errors

Day One Slide 57

Take a BREAK Sponge

• DURING THE BREAK, FIND TWO PEOPLE WITH WHOM YOU HAVE NOT YET SPOKEN.

• SHARE AN INSIGHT YOU’VE HAD THIS MORNING.

Day One Slide 59

Modeling Lesson 17

Differentiate instruction to meet your students’ needs.

Anticipatory Set

What is repeated addition?

Generalizing Rates

Chapter 4

Lesson 17

Understand that multiplication by rates and ratios can be used to transform an input into an output. Find the third when given two of the following: the input, the rate or ratio, the output.

Understand ratios as generalized rates, i.e. rates for which units have not been specified.

Objectives

• Generalize the concepts of rates to ratios.

• Use ratios to solve problems involving proportion relationships.

a. 2 × a = 14

b. 5 × b = 45

c. 64 = c × c

d. 27 = 9 × d

2. There are 3 oaks for every 5 pines. If there are 10 pines, how many oaks are there?

3. Find the value of g on the following number line.FROM pines TO oaks

5 pines + 5 pines

3 oaks + 3 oaks

Get Your Brain In Gear

Anticipatory Set

“In the previous couple of lessons we used rates to convert an input quantity measured in one unit to an output quantity measured in a different unit. In this lesson we will discuss this input-output relationship in more general terms.”

Prior Learning Today’s ObjectivesIntroductory Paragraph

When multiply by a rate to convert units, the rate is acting like a function input-output machine.

It takes in a quantity of input units and produces a quantity of output units.

If 3 apples cost 2 dollars, we write the rate from dollars to apples like this:

The store that sells these apples functions as a conversion machine. It takes in dollars and outputs apples.

If we put 2 dollars into the machine, the machine will produce 3 apples.

This more general machine takes in 2 “objects” and produces 3 “objects”:

We don’t need to specify the units:

Without units, this is a ratio. This ratio works like a rate, to turn 2 objects into 3 objects.

We call it a “3 for 2” ratio.

Let’s use the “3 for 2” ratio as an input-output machine.

We can use anything as the input, as long as we are consistent about what we mean by a “thing”.

If we put in 2 bananas, we’ll get out 3 bananas:If we put in 2 donuts, we’ll get out 3 donuts.

1. Find the ratio for each input-output machine:

Show the rope’s length as h + h so that there will be 2 parts, each of length h.

What would happen if we put the following piece of rope into the machine?

We can split the rope into 2 equal lengths:

The output rope is 3 lengths of +h, or h + h + h.

What will the output be?

Let’s use point 6 on the number line as an input to the 3-for-2 ratio machine:

First, create an expression for 6 using 2 equal jumps:What will the output be?The 3-for-2 ratio transforms point 6 into point 9. 3 + 3 + 3

Let’s try a different ratio to transform point 6 on the number line:

Now we put it into the 1-for-3 ratio machine. What is the output?The 1-for-3 ratio produces 2 for an input of 6. Create an expression for 6. How many equal jumps?

Day One Slide 75

The ___ equal jumps of ___ come out.

The ratio is ___ for ___

So, ___ needs to be made into ___ equal jumps.

The ___ equal jumps of ___ go in.

Monitor Students’ Understanding

2. Find the output for each machine:

Show your process.

A: Explain to B how you did (2b.).

B: Explain to A how you did (2c.).

3. Divide 24 into 6 equal parts by solving for z:

24 = 6 z

4. Divide 20 into 4 equal parts by solving for t:

20 = 4 t

5. Divide 16 into 2 equal parts by solving for k:

16 = 2 k

Check for UnderstandingWhat if students do fine with the previous problems, but

don’t know how to do the following?

6. Find the output of each machine:

How does this Check for Understanding #6 relate to Check for Understanding #3- #5?

Day One Slide 80

Scaffolding

6 ? 24 5 ?

1. What ratio was used in this machine?

Multiple Choice Practice

A student made 2 mistakes below. Find and correct each mistake.

Find the Errors

Teaching strategies build students’ confidence and competence.

Students build their understanding.

Students practice with feedback.

Students verbalize their understanding.

Review frequently and over time.

Monitor understanding and adjust instruction.

Direct Instructi

Lesson Slideshows

Hands-on Activities

ST Math Games

Variety

“We found that students who wrote journal entries on topics related to specific test questions were more likely to correctly answer those objective test questions than students who did not write on the topic.” C. A. Croxton, R. C. Berger*

Teaching Tips: Rate Word Problems

3 dollars 2 pens

2 pens 3 dollars

How many pens can I buy with $12?FROM _______ TO ________?

2 pens12 dollars =

3 dollars

Day One Slide 87

Content Experience: Rates

Rates.exe

Day One Slide 89

Day One Slide 90

Take a LUNCH Sponge

• DURING LUNCH, DISCUSS ONE STRATEGY YOU HAVE FOR MANAGING HOMEWORK.

• GIVE A COMPLIMENT TO ONE OTHER PARTICIPANT.

Day One Slide 91

Put it all together for student success.

Target instruction onkey concepts and big ideas.

Day One Slide 94

Day One Slide 95

Use Diagnostic assessments to customize pacing and instruction.

Chapters: 3, 5, 6, 8, 9, or 10

Day One Slide 96

Preview games through the teacher console.

www.mindresearch.net

Username: minltestPassword: algebra

Practice by accessing games, 4-5 version

Algebra ReadinessChapter 4: Rates and RatiosGame: Ratio Transform

Visualization Progression through Levels

Description:

Correct choice:

Error Depiction:

Level 1:

Level 2:

Level 3:

Level4:

Level 5:

Record the progression through the levels of the game.

Create connections; bring the games to classroom.

Day One Slide 102

Class Level Reports provide formative assessments.

Use reports to design interventions.

Day One Slide 105

Facilitating, step oneSTEP ONE

Uncover the Thinking

• What have you tried?• What happened then?• Why did you ______?

Day One Slide 106

Facilitating, step twoSTEP TWO

Examine the Animation• What is happening?• What did you notice? What else do you

notice?• When you clicked _____, what

happened?

Day One Slide 107

Facilitating, step threeSTEP THREE

Apply the Hypothesis

• What do you think will happen?• How will this work on this problem?• What steps will you take?• How did you get that answer?• How did you decide that was correct?

Day One Slide 108

Create an action plan that motivates.

Day One Slide 110

Survey and Evaluation

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Please participate in our survey and evaluation at:

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Day One Slide 113

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