Math is Exploring and Thinking
I can make sense of a problem, plan a strategy, and use fractions of a whole number to compare quantities.
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🎯 Content Objective / Objetivo de contenido
I can make sense of a problem, plan a strategy, and use fractions of a whole number to compare quantities.
Today's Flow
Total pacing: ~45 min · Progress bar at top tracks your place
LAUNCH
⏱ ~10 min
⏱️ 3 MIN · THINK-PAIR-SHARE
The book decomposes 105.76 as one example: 100 + 5.76. Why must the parts of a decomposition always rebuild the original number exactly?
Check for Understanding #1
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Unit 1 · Lesson 2
Doing math is exploring: you make sense of a problem, plan a strategy, check your progress, and try new paths when you hit a dead end. Today's problems — walking distances and skyscraper heights — reward thinkers, not just calculators.
Concept Launch
💡 Make sense, plan, check
The book opens with two warm-ups: about how many peas are in the picture, and decompose 105.76 five different ways. Both are about thinking before computing.
Math Practices: Sense Making & Strategy Planning. 1. Identify known quantities and the unknown question. 2. Choose a strategy (draw a model, write an equation, or build a table). 3. Evaluate progress and pivot to an alternate approach if needed
Check for Understanding #2
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Now you try
VOCABULARY
⏱ ~8 min
| Term / Término | Meaning / Significado | Example / Ejemplo | Visual |
|---|---|---|---|
| quantity cantidad |
An amount in a problem — a number with a meaning attached, like 540 meters. Una cantidad en un problema: un número con un significado, como 540 metros. |
The heights of the buildings are the quantities in the problem. | 540 meters — the number 540 AND what it measures. |
| relationship relación |
How two quantities compare or connect to each other. Cómo se comparan o se conectan dos cantidades entre sí. |
Deon walked a shorter distance than Miguel. | 1 hour → 48 km, 2 hours → 96 km: as one grows, so does the other. |
| strategy estrategia |
A plan for how to solve a problem before you start computing. Un plan para resolver un problema antes de empezar a calcular. |
We can think about how the quantities in the problem relate. | “I will draw a tape diagram first, then divide” — chosen before computing. |
| reasonable razonable |
A solution that makes sense in the context of the problem. Una solución que tiene sentido en el contexto del problema. |
A building that is 5/6 of 540 meters must be less than 540 meters. | 5/6 of 540 must be under 540, so 450 is reasonable and 650 is not. |
| persevere perseverar |
To keep working on a problem, checking your progress and adjusting your plan when you get stuck. Seguir trabajando en un problema, revisando tu progreso y ajustando tu plan cuando te atascas. |
When you hit a dead end, you try another strategy instead of quitting. | Stuck on the tram problem → try a tape diagram instead → keep going. |
Which Word Fits?
A number with a meaning attached, like 540 meters, is a ___.
Use It In a Sentence
Check for Understanding #3
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Turn & Talk — Launch
The book decomposes 105.76 as one example: 100 + 5.76. Why must the parts of a decomposition always rebuild the original number exactly?
👂 Listen For
A strong answer explains that each digit must land in its correct place value so the parts add back to exactly 105.76. A weak answer just says '100 and 5.76 are parts' without checking they sum correctly.
Extend: Find a decomposition of 105.76 that is different from 100 + 5.76, and prove it also sums to 105.76.
EXPLORE & PRACTICE
⏱ ~18 min
Visual Modeling Workspace
Use the drawing tray below to annotate the visual model. Teacher: say "Click to reveal" on key steps.
Explore Activity
The problem: Deon, Miguel, and Evelyn each walked a different distance through the neighborhood today. Deon walked a shorter distance than Miguel. Evelyn walked a longer distance than Miguel. Nobody wrote down the exact number of miles anyone walked. — Making sense of a problem starts with separating what you know from what you do not. Sort each card, then tell a partner: can you still figure out who walked the shortest distance, even with no exact numbers?
✍️ Explore Discourse
Pick 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?
Whiteboard Moment
Show your work clearly. Be ready to explain your thinking to a partner.
Turn & Talk — Explore
You sorted 'Deon walked a shorter distance than Miguel' as something we KNOW, even without exact numbers. How can a relationship like 'shorter than' be knowledge, if we don't know the actual distance?
👂 Listen For
Listen for students distinguishing between a relationship (comparison) and an exact value — a relationship is still usable information. A weak answer says we 'know nothing' because there are no numbers.
Extend: Can you determine who walked the SHORTEST distance among all three students using only the relationships given? Justify your order.
Practice Check A
The Rio de Janeiro building will be 9/10 as tall as One World Trade Center (540 meters). How tall will it be?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Practice Check B
The Tokyo building will be 6/5 as tall as One World Trade Center (540 meters). How tall will it be?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Ratio Table Builder
Fill the ratio table. Each row must be equivalent.
| Factor | A | B |
|---|---|---|
| ×1 | ||
| ×2 | ||
| ×3 |
✍️ Justify Your Thinking
Each new building is a fraction of One World Trade Center's 540 meters. Sort each plan without computing.
A classmate turned in the work below. One step has a mistake. Read every step, find it, name it, and fix it.
No worked steps provided for this lesson.
Choose ONE option to show what you know — then do it in the workspace below.
Use evidence from today's lesson to complete each frame.
Today's key idea is: "Math Practices: Sense Making & Strategy Planning. 1. Identify known quantities and the unknown question. 2. Choose a strategy (draw a model, write an equation, or build a table). 3. Evaluate progress and pivot to an alternate approach if needed" — and it works because ___.
Because quantity means ___, but a tricky part is ___, so I have to ___.
I can justify my method by showing ___, which proves my answer makes sense.
I can prove my answer is correct by ___, using relationship to check my work.
✍️ TWR · WRITE 3 SENTENCES · 7 MIN
Math Practices: Sense Making & Strategy Planning. 1. Identify known quantities and the unknown question. 2. Choose a strategy (draw a model, write an equation, or build a table). 3. Evaluate progress and pivot to an alternate approach if needed because ___
Math Practices: Sense Making & Strategy Planning. 1. Identify known quantities and the unknown question. 2. Choose a strategy (draw a model, write an equation, or build a table). 3. Evaluate progress and pivot to an alternate approach if needed but ___
Math Practices: Sense Making & Strategy Planning. 1. Identify known quantities and the unknown question. 2. Choose a strategy (draw a model, write an equation, or build a table). 3. Evaluate progress and pivot to an alternate approach if needed so ___
🌱 TWR · GROW THE KERNEL · 6 MIN
Answer these to add detail
Sentence starters (tap to use)
Student Workspace
The problem: Deon, Miguel, and Evelyn each walked a different distance through the neighborhood today. Deon walked a shorter distance than Miguel. Evelyn walked a longer distance than Miguel. Nobody wrote down the exact number of miles anyone walked. — Making sense of a problem starts with separating what you know from what you do not. Sort each card, then tell a partner: can you still figure out who walked the shortest distance, even with no exact numbers?
| Column A | Column B |
|---|---|
✏️ Sketch Your Strategy
Differentiation Paths
Step-by-step with a worked model and sentence frames.
Which of these is a correct way to decompose 105.76?
The Rio de Janeiro building will be 9/10 as tall as One World Trade Center (540 meters). How tall will it be?
Which set shows FIVE different correct decompositions of 105.76?
Partner Activity
Work with your partner on the practice problems at your differentiation path level. Explain each step using math vocabulary.
Check for Understanding #4
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Real-World Connection
🌍 Math in the Wild
Deon walked a shorter distance than Miguel, and Evelyn walked a longer distance than Miguel. You never learn the exact distances — but the relationships still tell you a lot.
✍️ Connection Reasoning
How can you use the relationships among the distances to choose possible values for how far Deon, Miguel, and Evelyn walked?
Pick a distance for Miguel. Then choose a distance for Deon that is ___ than Miguel's and a distance for Evelyn that is ___ than Miguel's, and explain how you ___ that your three values work.
Turn & Talk — Connect
If you pick a distance for Miguel, how do you choose distances for Deon and Evelyn that stay true to the relationships in the problem?
👂 Listen For
A strong answer picks a value for Miguel first, then chooses Deon's and Evelyn's values so the inequalities still hold, and explains the check. A weak answer picks three numbers with no verification against the relationships.
Extend: Could two different students pick completely different sets of three distances and BOTH be correct? Explain why or why not.
Summarize: Math is Exploring and Thinking
Math Practices: Sense Making & Strategy Planning. 1. Identify known quantities and the unknown question. 2. Choose a strategy (draw a model, write an equation, or build a table). 3. Evaluate progress and pivot to an alternate approach if needed
In your own words:
CLOSURE & REFLECT
⏱ ~8 min
Today I learned that ___ because ___.
One thing I am still not sure about is ___.
Which statement best shows what this lesson means by "math is exploring and thinking"?
Bonus Exit Check
Without calculating, which new building must be TALLER than One World Trade Center, and why?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Reflection & Self-Assessment
Continue Learning
Launch the Full Interactive Activity
Students continue practice in the HTML lesson engine with auto-check, hints, and differentiation.
Family Connection
Share tonight's family homework and discuss one vocabulary word at home.
Open Family Homework ↗Teacher Notes
⏱️ Pacing Guide
- Launch & vocab: 12 min
- I Do / We Do / You Do: 15 min
- Explore & practice: 15 min
- Connect & closure: 8 min
Total: ~45 min
🎯 Listen For · Common Errors
• A strong answer explains that each digit must land in its correct place value so the parts add back to exactly 105.76. A weak answer just says '100 and 5.76 are parts' without checking they sum correctly.
• Listen for students distinguishing between a relationship (comparison) and an exact value — a relationship is still usable information. A weak answer says we 'know nothing' because there are no numbers.
• A strong answer picks a value for Miguel first, then chooses Deon's and Evelyn's values so the inequalities still hold, and explains the check. A weak answer picks three numbers with no verification against the relationships.
• Listen for students naming a specific strategy (checking place value, adding the parts back together) rather than 'I just knew it.' A weak answer skips the verification step.
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.
Answer Key (Teacher Appendix)
Hide this slide during presentation or move to the end of your copy.
✓ Practice 1: 486 meters — 9/10 × 540: one tenth of 540 is 54, and 9 tenths is 9 × 54 = 486 meters.
✓ Practice 2: 648 meters — 6/5 × 540: one fifth of 540 is 108, and 6 fifths is 6 × 108 = 648 meters — taller than 540, because 6/5 is greater than 1.
✓ Practice 3: Tokyo, because 6/5 is greater than 1 — Multiplying 540 by a fraction greater than 1 gives more than 540. Only 6/5 is greater than 1; 5/6 and 9/10 are both less than 1.
✓ Practice 4: The answer is unreasonable — 5/6 of 540 must be LESS than 540 — 5/6 is less than 1, so 5/6 of 540 must land below 540. Getting 650 means the plan needs adjusting — which is exactly what checking is for.
✓ Exit ticket: We make sense of problems, plan a strategy, check our progress, and try other strategies at dead ends — The summary names the whole cycle: make sense of the problem, develop a plan, check progress and adjust, and make sense of the solution in context.