Grade 6 Mathematics · Pre-UnitStandard 5.NF.B.4

Lesson 1.2Math is Exploring and Thinking

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can make sense of a problem, plan a strategy, and use fractions of a whole number to compare quantities.

Start hereWords for this lesson

Read each word before you begin. Say it out loud.

WordWhat it meansExample
quantitySpanish: cantidad An amount in a problem — a number with a meaning attached, like 540 meters. 540 meters — the number 540 AND what it measures.
relationshipSpanish: relación How two quantities compare or connect to each other. 1 hour → 48 km, 2 hours → 96 km: as one grows, so does the other.
strategySpanish: estrategia A plan for how to solve a problem before you start computing. “I will draw a tape diagram first, then divide” — chosen before computing.
reasonableSpanish: razonable A solution that makes sense in the context of the problem. 5/6 of 540 must be under 540, so 450 is reasonable and 650 is not.
persevereSpanish: perseverar To keep working on a problem, checking your progress and adjusting your plan when you get stuck. Stuck on the tram problem → try a tape diagram instead → keep going.

How it worksWorked example

These numbers are not on your problems. The steps are. Follow them with your own numbers.

A bowl is full of peas — far too many to count one at a time. About how many peas are in the bowl?

  1. I do not count every pea — I look for a group I CAN count, like one small cluster of about 10.
  2. Then I ask how many of those clusters would cover the whole bowl. I count about 12.
  3. 12 clusters of about 10 is 12 × 10 = 120, so my estimate is about 120 peas — a reasoned answer, close enough to be useful.

Now you trySame steps, your turn

The steps are the same as the worked example. The numbers are yours. Do the work.

105.76 = 100 + 5.76.

Answer:

Say it and write itSentence starters

Finish each sentence out loud with a partner. Then use them in your writing.

  • To know , I first need , because depends on it.
  • I know that .
  • I don't know .
  • When I'm stuck, I can .

Word bank quantityrelationshipstrategyreasonablepersevere

Watch outA common mistake

Expecting a fraction of a number to be bigger than the number — 5/6 × 540 must be LESS than 540 because 5/6 is less than 1; only a fraction greater than 1, like 6/5, makes the product bigger.

Grade 6 Mathematics · Pre-UnitStandard 5.NF.B.4

Lesson 1.2Math is Exploring and Thinking

Version ASupported practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can make sense of a problem, plan a strategy, and use fractions of a whole number to compare quantities.

Part 1Apply itreal-world problems
  1. 1Circle the letter of the best answer. Show how you know.

    Which of these is a correct way to decompose 105.76?

    1. A10 + 5.76
    2. B100 + 5.76
    3. C100 + 5.076
    4. D105 + 7.6

    Hint Add each option and compare to 105.76.

    Show your work
  2. 2Circle the letter of the best answer. Show how you know.

    Which is another correct decomposition of 105.76?

    1. A106 − 0.34
    2. B110 − 4.24
    3. C110 − 5.24
    4. D110 + 4.24

    Hint Start at 110. How far down is 105.76?

    Show your work
  3. 3Circle the letter of the best answer. Show how you know.

    Deon walked a shorter distance than Miguel, and Evelyn walked a longer distance than Miguel. Who walked the shortest distance?

    1. ADeon
    2. BMiguel
    3. CEvelyn
    4. DIt cannot be determined

    Hint Put Miguel in the middle and place the other two around him.

    Show your work
  4. 4Circle the letter of the best answer. Show how you know.

    The London building will be 5/6 as tall as One World Trade Center, which is 540 meters. How tall will the London building be?

    1. A90 meters
    2. B450 meters
    3. C534 meters
    4. D648 meters

    Hint First find one sixth of 540.

    Rewrite each fraction
    Work it out
    Simplify
    1. 1Same size piecesCommon denominator first.
    2. 2OperateOnly then add or subtract.
    3. 3SimplifyDivide out common factors.
Part 2Explain and correctreasoning
  1. 5Answer in complete sentences.

    In the Comparing Walks problem, name one thing you KNOW and one thing you DON'T know.

    Sentence starter I know that ___ .
    I don't know ___ .
    Write your answer
  2. 6Answer in complete sentences.

    What can you do if you're stuck on a problem?

    Sentence starter When I'm stuck, I can ___ .
    Another strategy I could try is ___ .
    Write your answer

Explain your thinkingPick one card you put under 'We DON'T know this yet.' What would you have to figure out first to move it to the known column, and which quantity does it depend on?

Sentence starter To know ___, I first need ___, because ___ depends on it.

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Pre-UnitStandard 5.NF.B.4

Lesson 1.2Math is Exploring and Thinking

Version BCore practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can make sense of a problem, plan a strategy, and use fractions of a whole number to compare quantities.

Part 1Understand the ideamodels and meaning
  1. 1Circle the letter of the best answer. Show how you know.

    You calculate 5/6 × 540 and get 650 meters. What does the checking step of your plan tell you?

    1. AThe answer is fine because it is close to 540
    2. BThe answer is fine because 650 is a bigger number than 5/6
    3. CThe answer is unreasonable — 5/6 of 540 must be LESS than 540
    4. DChecking is only needed when the problem asks for it
    Show your work
Part 2Apply itreal-world problems
  1. 2Circle the letter of the best answer. Show how you know.

    The Rio de Janeiro building will be 9/10 as tall as One World Trade Center (540 meters). How tall will it be?

    1. A54 meters
    2. B486 meters
    3. C594 meters
    4. D4,860 meters
    Rewrite each fraction
    Work it out
    Simplify
  2. 3Circle the letter of the best answer. Show how you know.

    The Tokyo building will be 6/5 as tall as One World Trade Center (540 meters). How tall will it be?

    1. A108 meters
    2. B432 meters
    3. C450 meters
    4. D648 meters
    Rewrite each fraction
    Work it out
    Simplify
  3. 4Circle the letter of the best answer. Show how you know.

    Without calculating, which new building must be TALLER than One World Trade Center, and why?

    1. ALondon, because 5/6 has a 6 in it
    2. BTokyo, because 6/5 is greater than 1
    3. CRio de Janeiro, because 9/10 is close to 1
    4. DNone — a fraction of 540 is always less than 540
    Show your work
Part 3Explain and correctreasoning
  1. 5Answer in complete sentences.

    Miguel walked 2 miles. Deon walked less than Miguel, and Evelyn walked more than Miguel. Choose possible distances for Deon and Evelyn and explain how you checked both relationships.

    Write your answer
  2. 6Answer in complete sentences.

    The lesson says we check on our progress toward a solution and adjust our plan as needed. Describe one way YOU can check on your progress while solving a problem.

    Write your answer

Explain your thinkingPick one card you put under 'We DON'T know this yet.' What would you have to figure out first to move it to the known column, and which quantity does it depend on?

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Pre-UnitStandard 5.NF.B.4

Lesson 1.2Math is Exploring and Thinking

ChallengeExtension

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can make sense of a problem, plan a strategy, and use fractions of a whole number to compare quantities.

Part 1Understand the ideamodels and meaning
  1. 1Write the letter of the matching item on each line.

    Match each fraction of 540 to a shortcut that computes it.

    1. 9/10 of 540
    2. 6/5 of 540
    3. 5/6 of 540
    4. 1/2 of 540
    • A540 − 54
    • B540 + 108
    • C540 − 90
    • D540 ÷ 2
Part 2Apply itreal-world problems
  1. 2Circle the letter of the best answer. Show how you know.

    Which set shows FIVE different correct decompositions of 105.76?

    1. A100 + 5.76, 105 + 0.76, 110 − 4.24, 52.88 × 2, 211.52 ÷ 2
    2. B100 + 5.76, 105 + 7.6, 110 − 4.24, 52.88 × 2, 211.52 ÷ 2
    3. C100 + 5.76, 105 + 0.76, 110 + 4.24, 52.88 × 2, 211.52 ÷ 2
    4. D100 + 5.76, 105 + 0.76, 110 − 4.24, 52.88 × 3, 211.52 ÷ 2
    Show your work
  2. 3Circle the letter of the best answer. Show how you know.

    Without computing the exact heights, order the three new buildings from shortest to tallest. (London: 5/6 of 540 m, Rio de Janeiro: 9/10 of 540 m, Tokyo: 6/5 of 540 m.)

    1. ARio de Janeiro, London, Tokyo
    2. BTokyo, Rio de Janeiro, London
    3. CLondon, Tokyo, Rio de Janeiro
    4. DLondon, Rio de Janeiro, Tokyo
    Show your work
Part 3Explain and correctreasoning
  1. 4Answer in complete sentences.

    A classmate says, "In math, only the final answer matters." Using this lesson, write what you would say back.

    Write your answer
  2. 5Answer in complete sentences.

    Write your own Comparing Walks problem: describe three people using only relationships (shorter than, longer than), so a reader can put them in order without any exact distances. Then give one set of values that fits.

    Write your answer
  3. 6Answer in complete sentences.

    Apply — City Blocks. A city block has 12 buildings. The block to the north is planned to hold 2/3 as many buildings, and the block to the east is planned to hold 3/2 as many. WITHOUT computing, say which block will hold the most buildings and how you know. Then find the number of buildings on each block and check that your answers make sense.

    Write your answer

Explain your thinkingPick one card you put under 'We DON'T know this yet.' What would you have to figure out first to move it to the known column, and which quantity does it depend on?

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help