How to run this lesson. Send the pre-work a day ahead. Open with goals (5 min), then ratios (12), percents (14), averages & data (14), probability & rates (8), the Vocabulary Lab (7), and a wrap that assigns the next-two-days work. The dark navy notes are for the instructor only.
The frame for today
One line to open: “These problems are reading problems with numbers — translate carefully and the math is easy.” Today is the last math domain; afterward we turn to Reading & Writing.
00 Before you begin
Pre-work handout
Five minutes of warm-up makes the lesson land faster.
Data & Problem-Solving — Pre-work
Complete on paper before our session · about 10 minutes
Warm-up
1. What is 10% of 50? ______
2. The average of 4 and 6 is ______.
3. If 2 pens cost $3, one pen costs ______.
4. A bag has 1 red and 3 blue marbles. P(red) = ______.
Answer key (instructor)
1 · 5 2 · 5 3 · $1.50 4 · 1/4
02 Ratios & proportions
Find the unit, then scale
Most ratio problems crack open the moment you find the value of one unit, then multiply.
Unit rate1
If 3 pens cost $4.50, how much do 7 pens cost?
Why B
One pen costs $4.50 / 3 = $1.50. Seven pens: 7 × $1.50 = $10.50.Solve it another way
True shortcut Divide to the price of one, then multiply. “Per” means divide.
Proportion2
The ratio of cats to dogs in a shelter is 2 : 3. If there are 12 cats, how many dogs?
Why B
2 : 3 means dogs = (3/2) × cats. With 12 cats: (3/2)(12) = 18.03 Percents
Translate the words to math
“Of” means multiply. Percent change is the difference over the original. An increase multiplies by (1 + r); a decrease multiplies by (1 − r).
The percent toolbox
“of” = ×
30% of 90 means 0.30 × 90 = 27.
Percent change
(new − old) ÷ old, then × 100.
One-step up/down
Up r%: ×(1 + r). Down r%: ×(1 − r).
Percent of3
What is 30% of 90?
Why B
30% of 90 = 0.30 × 90 = 27.Percent change4
A price rises from $40 to $50. What is the percent increase?
Why B
Change ÷ original = (50 − 40) / 40 = 10/40 = 25%.04 Averages & data
Mean, median, and reading data
Mean is the sum divided by the count. Median is the middle value once the data is in order.
Mean5
The average (mean) of 4, 8, and x is 7. What is x?
Why B
Mean = sum / count, so (4 + 8 + x)/3 = 7 → 12 + x = 21 → x = 9.Solve it another way
True shortcut Sum = mean × count. Here the three numbers must total 21.
Median6
What is the median of 3, 7, 9, 12, and 20?
Why B
In order, the middle of five values is the 3rd one → 9. (10.2 is the mean — a trap.)05 Probability & rates
Favorable over total
Probability is the number of favorable outcomes divided by the total. A rate is just a ratio with units, like miles per hour.
Probability7
A bag holds 3 red and 5 blue marbles. What is the probability of drawing a red marble?
Why B
Favorable / total = 3 red / 8 total = 3/8.Rate8
A car travels 150 miles in 3 hours. What is its average speed?
Why B
Speed = distance / time = 150 / 3 = 50 mph.Translate, don’t panic
These are reading problems with numbers. Have the student underline the question actually asked and label units before computing. The classic trap is reporting the mean when the median is asked (or the discount instead of the final price) — slow the last step down.
06 Vocabulary Lab
This week’s words — think in synonyms and antonyms
Tap a card to flip it. Learn each word next to its opposite — that’s how the test frames them.
Quick check
Synonym9
Which word is closest in meaning to negligible?
Why B
Negligible = too small to matter, like insignificant.Antonym10
Which word is most nearly opposite to approximate?
Why C
Approximate = roughly right; its opposite is exact.Word in context11
Desert temperatures ______ sharply, soaring by day and plunging at night.
Why A
Fluctuate = rise and fall irregularly — exactly the day/night swing.07 Wrap-up
Three things to carry out the door
Say them out loud — that’s how they stick.
Find the unit first
Price of one, then multiply — most ratio problems open right up.
“Of” means ×
And percent change is difference over the original.
Read the last step
Mean vs median, discount vs final price — answer what was asked.
Close the loop
Have the student set one target for the mid-week. Confirm the two lesson days, point them to the workbook, and preview that Lesson 6 turns to Reading — craft, structure, and evidence. The math front-load is complete.
08 Keep the fire lit
Next-two-days handout
Short, daily, cumulative.
Data & Problem-Solving — Work for the Next Two Days
About 30 minutes a day · label your units
Day 1 · Ratios & percents
Translate to math
1. 5 apples cost $3.00. Cost of 8 apples? ______
2. What is 40% of 75? ______
3. A price falls from $80 to $60. Percent decrease? ______
4. Ratio 4 : 5. If there are 16 of the first, how many of the second? ______
5. Increase 50 by 20%. ______
Day 2 · Averages, probability, rates
Read the last step
1. Mean of 6, 10, and 14? ______
2. Median of 2, 5, 9, and 11? ______
3. A bag has 4 green and 6 yellow. P(green)? ______
4. 240 miles in 4 hours. Average speed? ______
Answer key (instructor)
Day 1: 1 · $4.80 2 · 30 3 · 25% 4 · 20 5 · 60
Day 2: 1 · 10 2 · 7 3 · 2/5 4 · 60 mph
Day 2: 1 · 10 2 · 7 3 · 2/5 4 · 60 mph
This sets up Lesson 6. The careful reading you just used on word problems is the same skill that powers evidence-based reading next week.