Learn the structure
Book pages 49–52
- Explore and Share
- Visual Learning examples
- Selected check problems
Topic 2 • Analyze and Use Proportional Relationships
Ratios, Rates,
and Unit Rates
Open your student book to Lesson 2-1, page 49. Keep your math notebook beside it for every calculation.
The book gives us the problem. Your notebook shows the thinking.
Before we begin
Your two-day book plan
Apply and prove it
Book pages 53–54
- Selected practice problems
- Four proof problems checked for credit
- One optional stretch problem
You are not completing every problem on every page.
Do Now • Notebook • 5 minutes
What do you remember about ratios?
1Simplify
6 red : 9 blue
2Equivalent?
35 and 1220
3Unit rate
18 pages in 3 minutes
4Better buy
A: 2 notebooks for $8
B: 5 notebooks for $15
1) 2:3 2) Yes 3) 6 pages per minute 4) B, because $3 per notebook is less than $4.
Learning intention and success criteria
What are we learning?
Learning intention
We are learning to use ratios, rates, and unit rates to compare quantities and solve multi-step problems.
I will know I am successful when I can…
- identify the quantities and units;
- divide to find a rate “for 1”;
- use that rate to compare or solve;
- show and label every step.
TEKS 7.4B Calculate unit rates from mathematical and real-world rates.
Explore and Share • Book page 49
More made shots does not automatically mean a better rate.
Elizabeth
9 made25 attempts
Use a decimal, percent, or equivalent ratio.
Alex
8 made20 attempts
Use the same type of comparison.
Who has the better free-throw record? What model proves it?
Elizabeth: 9 ÷ 25 = 0.36, or 36%. Alex: 8 ÷ 20 = 0.40, or 40%. Alex has the better rate.
The structure
Ratio → Rate → Unit Rate
Compares quantities
3 red cards for every 5 blue cards.
3 : 5
Uses different units
180 miles in 3 hours.
180 miles3 hours
Has a denominator of 1
Divide both quantities by 3.
60 miles1 hour
A unit rate answers: “How much for one?”
Visual Learning • Book page 50
I Do: Compare hourly earnings
Dan
$141.759 hours = $15.751 hour
Nathan
$81.255 hours = $16.251 hour
Nathan earns $0.50 more per hour.
- Identify what you are finding: dollars per hour.
- Put dollars on top and hours on the bottom.
- Divide and label the unit rate.
Try It • Book page 50
We Do: Jennifer’s paycheck
Given
$242.25 for 15 hours
Find the amount earned for 1 hour.
Set it up
$242.2515 hours = ?1 hour
What operation makes the denominator 1?
$242.25 ÷ 15 = $16.15 per hour. The reverse unit rate is approximately 0.062 hour per dollar.
Use Unit Rates • Book page 51
I Do: Is 128 ounces of dog food enough?
Buster
20.5 oz2 days = 10.25 oz/day
10.25 × 7 = 71.75 oz
Roxy
22.5 oz3 days = 7.5 oz/day
7.5 × 7 = 52.5 oz
71.75 + 52.5 = 124.25 oz < 128 oz
Yes. There are 3.75 ounces left.
Try It • Book page 51
Which faucet fills 3 gallons faster?
Kitchen faucet
0.5 gal10 sec
Bathroom faucet
0.75 gal18 sec
Choose one comparison: gallons per second or seconds per gallon.
Kitchen: 0.05 gal/sec, so 3 gallons takes 60 sec. Bathroom: about 0.0417 gal/sec, so 3 gallons takes 72 sec. The kitchen faucet is faster.
Guided practice • Three problems only
Find, compare, and explain
1Find
4 pounds of oranges cost $10.
Cost per pound?
$10 ÷ 4 = $2.50 per pound.
2Compare
3 tickets for $21 or 5 tickets for $32.50?
Better buy?
$7 each versus $6.50 each. Five tickets are the better buy.
3Explain
126 pages in 3 hours or 180 pages in 4 hours?
Who reads faster?
42 pages/hour versus 45 pages/hour. The second reader is faster by 3 pages/hour.
Day 1 Book Work • Pages 49–52
Exactly what you are completing today
- p.49: Explore A and B
- p.50: Try It
- p.51: Try It
- p.52: #2 and #3
- p.52: #4 and #6
Show every setup in your notebook. A final answer without units is incomplete.
Lesson 2-1 • Day 2
Apply the Unit Rate
Open to page 53. Today you will solve selected problems and show independent learning.
Name the unit → divide → use the rate → label the answer.
Day 2 Reconnect
Which way should the division go?
If asked for…
dollars per pound
dollarspounds
If asked for…
miles per hour
mileshours
If asked for…
steps per mile
stepsmiles
The word after per belongs on the bottom.
Error analysis • Preview page 54, #14
Seven dozen is not seven pairs.
Incorrect reasoning
$103.32 ÷ 7 = $14.76
The contractor divided by 7 dozen but labeled the result “per pair.”
Correct reasoning
7 × 12 = 84 pairs
$103.32 ÷ 84 = $1.23
Before dividing, convert the denominator into the unit the question requests.
Day 2 Book Work • Pages 53–54
Must Do, Proof Problems, and Stretch
Complete all eight
p.53: #7, #8, #10, #12
p.54: #13, #14, #16, #17
Checked closely for credit
#7, #12, #14, and #16
Setup, visible work, units, and a labeled answer are required.
After the Must Do list
p.54, #15
Compare cost per ounce across three tuna packages.
Individual assignment. No help from a neighbor during the Proof Problems.
Independent expectations
What earns credit?
Setup
Write the ratio in the order requested.
Visible work
Show the division or equivalent-ratio steps.
Units
Label the unit rate and final answer.
Reasoning
Answer the question asked, not only the calculation.
If you are stuck: circle the two quantities, underline “per,” and identify what must equal 1.
Exit Ticket • Notebook • Work alone
Prove that you can find and use a unit rate.
1. Find
A car travels 72 miles in 1.5 hours.
What is its speed in miles per hour?
2. Compare
A: 12 ounces for $4.56
B: 18 ounces for $6.30
Which has the lower cost per ounce?
1) 72 ÷ 1.5 = 48 miles per hour. 2) A costs $0.38/oz; B costs $0.35/oz. Package B has the lower unit price.
Lesson 2-1 complete
Find the rate
for one.
Day 1
Pages 49–52
p.52: #2, #3, #4, #6
Day 2
Pages 53–54
#7, #8, #10, #12, #13, #14, #16, #17
Finish any incomplete Must Do problems. Proof Problems must include visible work and units.