Lesson 1.5Math is Finding Patterns
Start hereWords, worked example, and sentence starters
Learning target I can find patterns and pattern rules, use them to make generalizations, and check my solutions with a table of values.
Start hereWords for this lesson
Read each word before you begin. Say it out loud.
| Word | What it means | Example |
|---|---|---|
| patternSpanish: patrón | Something that repeats or changes in a predictable way. | 1, 3, 7, 15, 31 — each tower's steps follow a predictable jump. |
| pattern ruleSpanish: regla del patrón | The instruction that tells how to get from one number in a pattern to the next. | 2, 4, 6, 8 … the rule is “+ 2 every time”. |
| generalizationSpanish: generalización | A statement that is true for a whole pattern, made by noticing repeated calculations. | 1 → 3, 2 → 5, 3 → 7 … so n → 2n + 1, for ANY n. |
| table of valuesSpanish: tabla de valores | A table that lists each step of a pattern so you can check that a solution is reasonable. | discs 1, 2, 3 beside steps 1, 3, 7 — one row per step. |
| reasonablenessSpanish: sensatez del resultado | Whether an answer makes sense — checked as you work, with adjustments made when needed. | 5/6 of 540 should land BELOW 540, so 650 fails the check. |
How it worksWorked example
These numbers are not on your problems. The steps are. Follow them with your own numbers.
A change jar is full of pennies, nickels, dimes, and quarters worth $53.50 in total. How many of each coin could there be?
- Each penny has a value of $0.01, so 100 pennies have a value of $1.00.
- Each nickel has a value of $0.05, so 100 nickels have a value of $5.00. Each dime has a value of $0.10, so 100 dimes have a value of $10. Each quarter has a value of $0.25, so 100 quarters have a value of $25.
- I multiply the number of each coin by its value. And I notice a generalization: because the jar's total, $53.50, ends in 0, the number of pennies can only be a multiple of 5.
Now you trySame steps, your turn
The steps are the same as the worked example. The numbers are yours. Do the work.
Try 200 quarters: 200 × $0.25 = $50.00.
Say it and write itSentence starters
Finish each sentence out loud with a partner. Then use them in your writing.
- is only a one-time fact because ; it would be a pattern if .
- I noticed that .
- This helps because instead of counting one at a time, I can .
- I look for .
Word bank patternpattern rulegeneralizationtable of valuesreasonableness
Watch outA common mistake
Reporting only the points from the streak (20 or 30) and forgetting to add the points each player had before the streak began.
Lesson 1.5Math is Finding Patterns
Version ASupported practice
Learning target I can find patterns and pattern rules, use them to make generalizations, and check my solutions with a table of values.
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1Circle the letter of the best answer. Show how you know.
Each penny has a value of $0.01. What is the value of 100 pennies?
- A$0.10
- B$1.00
- C$10.00
- D$100.00
Hint A penny is one hundredth of a dollar.
Show your work -
2Circle the letter of the best answer. Show how you know.
Each quarter has a value of $0.25. Following the pattern, what is the value of 100 quarters?
- A$2.50
- B$25.00
- C$100.25
- D$250.00
Hint Multiplying by 100 moves the decimal point.
Show your work -
3Circle the letter of the best answer. Show how you know.
Player A scores a two-pointer on every shot. What is Player A's pattern rule?
- AAdd 2
- BMultiply by 2
- CAdd 3
- DAdd 18
Hint How much is one two-pointer worth?
Show your work -
4Circle the letter of the best answer. Show how you know.
Player B had 6 points, then made 10 three-pointers in a row. How many points does Player B have now?
- A9
- B16
- C30
- D36
Hint How many points do 10 three-pointers add?
Show your work
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5Answer in complete sentences.
Name a pattern you noticed in the coin values, and how it could help you count money faster.
Sentence starter I noticed that ___ .
This helps because instead of counting one at a time, I can ___ .Write your answer -
6Answer in complete sentences.
Where do you look for a pattern first when a problem feels too big to count?
Sentence starter I look for ___ .
A repeated calculation helps because ___ .Write your answer
Explain your thinkingTake a card you called 'A one-time fact.' What would have to keep happening for it to become a generalization instead — and how would you write that as a pattern rule?
Sentence starter ___ is only a one-time fact because ___; it would be a pattern if ___.
Lesson 1.5Math is Finding Patterns
Version BCore practice
Learning target I can find patterns and pattern rules, use them to make generalizations, and check my solutions with a table of values.
-
1Circle the letter of the best answer. Show how you know.
Player A had 18 points before making 10 two-pointers in a row. How many points does Player A have after the streak?
- A20
- B28
- C38
- D48
Show your work -
2Circle the letter of the best answer. Show how you know.
Akela's jar totals $53.50. Why can the number of pennies only be a multiple of 5?
- ABecause the total's cents end in 0, and only pennies make amounts that aren't multiples of 5 cents
- BBecause pennies always come in rolls of 5
- CBecause there are exactly 5 kinds of coins in the jar
- DBecause $53.50 is divisible by 5
Show your work -
3Circle the letter of the best answer. Show how you know.
Which combination of coins totals exactly $53.50?
- A100 quarters, 25 dimes, 15 nickels, 25 pennies
- B200 quarters, 25 dimes, 15 nickels, 25 pennies
- C200 quarters, 25 dimes, 15 nickels, 20 pennies
- D200 quarters, 30 dimes, 15 nickels, 25 pennies
Show your work -
4Circle the letter of the best answer. Show how you know.
How does a table of values check that Player A really scored 38 points?
- AIt rounds the answer to the nearest ten
- BIt replaces the pattern rule so you don't need one
- CIt proves the answer without doing any calculations
- DIt lists the score after every shot, so you can see the pattern land exactly on 38
Show your work
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5Answer in complete sentences.
Use the multiple-of-5 pattern to find a DIFFERENT combination of coins that totals $53.50. Show that it works.
Write your answer -
6Answer in complete sentences.
After the streaks, Player A has 38 and Player B has 36. Player B started 12 points behind but nearly caught up. What about the two pattern rules explains that?
Write your answer
Explain your thinkingTake a card you called 'A one-time fact.' What would have to keep happening for it to become a generalization instead — and how would you write that as a pattern rule?
Lesson 1.5Math is Finding Patterns
ChallengeExtension
Learning target I can find patterns and pattern rules, use them to make generalizations, and check my solutions with a table of values.
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1Write the letter of the matching item on each line.
Match each input-output table to its pattern rule.
- In: 1, 2, 3 → Out: 5, 10, 15
- In: 1, 2, 3 → Out: 3, 5, 7
- In: 2, 4, 6 → Out: 1, 2, 3
- In: 1, 2, 3 → Out: 2.5, 5, 7.5
- AMultiply by 5
- BDouble, then add 1
- CDivide by 2
- DMultiply by 2.5
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2Circle the letter of the best answer. Show how you know.
If both players kept their streaks going, after how many MORE shots would Player B tie Player A?
- A1 more shot
- BThey can never tie
- C2 more shots
- D10 more shots
Show your work -
3Circle the letter of the best answer. Show how you know.
A sei whale measures 4.5 meters at birth and grows about 2.5 centimeters each day. About how many days until it reaches 15 meters?
- AAbout 4.2 days
- BAbout 42 days
- CAbout 420 days
- DAbout 600 days
Show your work
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4Answer in complete sentences.
Write a generalization about ANY player who makes n shots in a row worth p points each, starting from s points. Then test it on Player A.
Write your answer -
5Answer in complete sentences.
The lesson says we 'evaluate the reasonableness of solutions as we work and make adjustments.' Describe a moment solving the coin-jar problem where you would stop and adjust.
Write your answer