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Suggested norms for talking math:

Give quiet think time before sharing ideas.
If someone is stuck, ask questions, don’t tell.
Try to think of different ways, not just one way.
We learn from our mistakes.
If everyone is stuck, notice and wonder together, take a break, or try to think of a similar problem you could do (this would be easier if…).
Brainstorm and write down your ideas together.

Launch

What do you notice?

What do you wonder?

Talking Math 6–8

Norms

www.illustrativemathematics.org CC-BY-NC

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Norms

Suggested norms for talking math:

Give quiet think time before sharing ideas.
If someone is stuck, ask questions, don’t tell.
Try to think of different ways, not just one way.
Mistakes grow your brain.
If everyone is stuck, notice and wonder together, take a break, or try to think of a similar problem you could do (this would be easier if…).
Brainstorm and write down your ideas together.

Elena is in 6th grade and she tries all the problems. She’s not sure how to do them all, but she plays around with some ideas. Why might Elena do that? What would be some good ways to help Elena if she gets frustrated?
Diego’s older brother sees him working on math and says, “Can I show you how you’re supposed to do it?” What could Diego say to his brother?
Jada says, “This problem should be easy. I feel stupid.” What could you say to Jada to encourage her to keep trying?

Talking Math 6–8

www.illustrativemathematics.org CC-BY-NC

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Normas sugeridas para hablar de matemáticas:

Dé tiempo para pensar en silencio antes de compartir ideas.
Si alguien está atascado, haga preguntas, no se lo diga.
Trate de pensar en diferentes formas, no solo en una.
Los errores hacen crecer tu cerebro.
Si todos están estancados, fíjese y pregúntense juntos, tómese un descanso o trate de pensar en un problema similar que podría resolver (esto sería más fácil si ...).
Haga una lluvia de ideas y escriba sus ideas juntos.

Introducción

¿Qué observas?

¿Qué te preguntas?

Normas

Talking Math 6–8

ES

www.illustrativemathematics.org CC-BY-NC

4 of 40

Normas

Normas sugeridas para hablar de matemáticas:

Dé tiempo para pensar en silencio antes de compartir ideas.
Si alguien está atascado, haga preguntas, no se lo diga.
Trate de pensar en diferentes formas, no solo en una.
Los errores hacen crecer tu cerebro.
Si todos están estancados, fíjese y pregúntense juntos, tómese un descanso o trate de pensar en un problema similar que podría resolver (esto sería más fácil si ...).
Haga una lluvia de ideas y escriba sus ideas juntos.

Elena está en sexto grado y trata todos los problemas. No está segura de cómo hacerlos todos, pero juega con algunas ideas. ¿Por qué Elena podría hacer eso? ¿Cuáles serían algunas buenas maneras de ayudar a Elena si se frustra?
El hermano mayor de Diego lo ve trabajando en matemáticas y dice: "¿Puedo mostrarle cómo se supone que debe hacerlo?" ¿Qué podría decirle Diego a su hermano?
Jada dice: “Este problema debería ser fácil. Me siento estupido." ¿Qué podrías decirle a Jada para alentarla a seguir intentándolo?

Talking Math 6–8

ES

www.illustrativemathematics.org CC-BY-NC

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Launch

What do you notice?

What do you wonder?

Talking Math 6–8

Week 1

www.illustrativemathematics.org CC-BY-NC

6 of 40

What is the area of polygon EFGH? Explain or show how you know.
Measure and record the horizontal and vertical lengths of each polygon. For example, in polygon EFGH, what is the length of HF and GE? How does the ratio of horizontal:vertical compare?
Each of these polygons is a scaled copy of the others. A scaled copy is a copy of a figure where every length in the original figure is multiplied by the same number. Name two pairs of corresponding angles. What can you say about the sizes of these angles?
Draw your own three shapes that are scaled copies of one another.

Talking Math 6–8

Week 1

www.illustrativemathematics.org CC-BY-NC

7 of 40

Introducción

¿Qué observas?

¿Qué te preguntas?

Semana 1

Talking Math 6–8

ES

www.illustrativemathematics.org CC-BY-NC

8 of 40

¿Cuál es el área del polígono EFGH? Explica o muestra cómo lo sabes.
Mide y registra las longitudes horizontal y vertical de cada polígono. Por ejemplo, en el polígono EFGH, ¿cuál es la longitud de HF y GE? ¿Cómo se compara la relación horizontal: vertical?
Cada uno de estos polígonos es una copia a escala de los demás. Una copia a escala es una copia de una figura donde cada longitud de la figura original se multiplica por el mismo número. Nombra dos pares de ángulos correspondientes. ¿Qué puedes decir sobre los tamaños de estos ángulos?
Dibuja tres formas que sean copias escaladas una de la otra.

Semana 1

Talking Math 6–8

ES

www.illustrativemathematics.org CC-BY-NC

9 of 40

Launch

What do you notice?

What do you wonder?

Talking Math 6–8

Week 2

www.illustrativemathematics.org CC-BY-NC

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Week 2

About how tall is the actual Willis Tower? About how tall is the actual Great Pyramid? Explain or show your reasoning.
The tallest mountain in the United States, Mount Denali in Alaska, is about 6,190 m tall. If this mountain were shown on the scale drawing, how would its height compare to the heights of the structures? Explain or show your reasoning.

What do you think the segment labeled “0” and “100 m” means?
If your school were added to the image, what would it look like?
If the tallest building where you live were added, what would it look like?

Talking Math 6–8

www.illustrativemathematics.org CC-BY-NC

Instructions:

Everyone try question A—give everyone 2-4 minutes of quiet think time and then start talking about your ideas.
Sentence/question starters for grown-ups and kids:

Tell me more about . . .
Are there more ways we can try?
Can you help me understand . . . ?
How do you see it? Can you show me with a drawing?

Keep going with any other problems that feel familiar and do-able.
Are you ready for more?

Reflect on what you did and how it went.
Connect with a classmate and ask them about their ideas.

Highlight below to see solutions:

A. Sample response: In the picture, the length from the 0 to the 100 represents 100 meters. The Great Pyramid is more than 100 m tall.

B. Sample response: My school is only one story tall, so you would barely be able to see it at this scale.

C. Sample response: The tallest building in my town is a 10 story hotel. It’s about 40 meters tall, so it would not even reach halfway up the height of the Great Pyramid.

D. The Willis Tower is a bit more than 500 m tall. It takes about 5 of the segment lengths to measure the height. The Great Pyramid is approximately 150 m. Its height is about 1 1/2 segments long.

E. Mount Denali will be far taller (about 7.5 times taller) than the Burj Khalifa. Sample explanations: I found out that the Burj Khalifa is about 830 m. Mount Denali is 6,190 m, and 6,190 divided by 830 is about 7.5, so the mountain will be about 7.5 times taller than the Burj Khalifa.

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Introducción

¿Qué observas?

¿Qué te preguntas?

Semana 2

Talking Math 6–8

ES

www.illustrativemathematics.org CC-BY-NC

12 of 40

Semana 2

¿Aproximadamente qué altura tiene la verdadera Torre Willis? ¿Aproximadamente qué altura tiene la verdadera Gran Pirámide? Expliquen o muestren su razonamiento.
La montaña más alta de los Estados Unidos, el monte Denali en Alaska, tiene unos 6.190 m de altura. Si esta montaña se mostrara en el dibujo a escala, ¿cómo se compararía su altura con las alturas de las estructuras? Expliquen o muestren su razonamiento.

¿Qué crees que significa el segmento etiquetado "0" y "100 m"?
Si tu escuela fuera agregada a la imagen, ¿cómo sería?
Si se añadiera el edificio más alto donde vives, ¿cómo sería?

Talking Math 6–8

ES

www.illustrativemathematics.org CC-BY-NC

Instructions:

Everyone try question A—give everyone 2-4 minutes of quiet think time and then start talking about your ideas.
Sentence/question starters for grown-ups and kids:

Tell me more about . . .
Are there more ways we can try?
Can you help me understand . . . ?
How do you see it? Can you show me with a drawing?

Keep going with any other problems that feel familiar and do-able.
Are you ready for more?

Reflect on what you did and how it went.
Connect with a classmate and ask them about their ideas.

Highlight below to see solutions:

A. Sample response: In the picture, the length from the 0 to the 100 represents 100 meters. The Great Pyramid is more than 100 m tall.

B. Sample response: My school is only one story tall, so you would barely be able to see it at this scale.

C. Sample response: The tallest building in my town is a 10 story hotel. It’s about 40 meters tall, so it would not even reach halfway up the height of the Great Pyramid.

D. The Willis Tower is a bit more than 500 m tall. It takes about 5 of the segment lengths to measure the height. The Great Pyramid is approximately 150 m. Its height is about 1 1/2 segments long.

E. Mount Denali will be far taller (about 7.5 times taller) than the Burj Khalifa. Sample explanations: I found out that the Burj Khalifa is about 830 m. Mount Denali is 6,190 m, and 6,190 divided by 830 is about 7.5, so the mountain will be about 7.5 times taller than the Burj Khalifa.

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Week 3

Talking Math 6–8

Launch

Here are some hangers with weights hanging on each side.

What do you notice?

What do you wonder?

www.illustrativemathematics.org CC-BY-NC

14 of 40

Week 3

Talking Math 6–8

For each hanger diagram, find:

one thing that must be true
one thing that could be true or false
one thing that cannot possibly be true

What do you think will happen to each hanger if another square is added to both sides of the hanger? What about a circle with weight x?
If 1 green triangle weighs 12.6 grams, what is the weight of 1 blue square?
Create your own hanger diagram for someone to solve. Use at least 3 shapes all with distinct weights.
We can use the equation y = 3x to describe the hanger on the right. What would the hanger for x + 3y = 2x + y look like?
If y = 3x, is the hanger for x + 3y = 2x + y balanced? Explain or show how you know.

www.illustrativemathematics.org CC-BY-NC

Instructions:

Everyone try question A—give everyone 2-4 minutes of quiet think time and then start talking about your ideas.
Sentence/question starters for grown-ups and kids:

Tell me more about . . .
Are there more ways we can try?
Can you help me understand . . . ?
How do you see it? Can you show me with a drawing?

Keep going with any other problems that feel familiar and do-able.
Are you ready for more?

Reflect on what you did and how it went.
Connect with a classmate and ask them about their ideas.

Highlight below to see solutions:

A. Sample response:

Left1. The triangle is heavier than the square.

Left2. The triangle could weigh 10 ounces and the square could weigh 6 ounces.

Left3. The square and the triangle weigh the same.

Right1. One triangle weighs the same as three squares.

Right2. The triangle weighs three pounds and each square weighs one pound.

Right3. One square is heavier than the triangle.

B. Sample response: I think the hangers will stay the same since equal weight is being added to each side.

C. 4.2 grams

D. Responses vary.

E. 1 square and 2 triangles on the left side of the hanger, 2 squares and 1 triangle on the right side.

F. No, the hanger is not balance. Sample response: The hanger for x + 3y = 2x + y has the weight of 10 x’s on the left but only 5 on the right, so it is not in balance.

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Semana 3

Introducción

Aquí hay algunas colgadores con pesas colgando a cada lado.

¿Qué observas?

¿Qué te preguntas?

Talking Math 6–8

ES

www.illustrativemathematics.org CC-BY-NC

16 of 40

Talking Math 6–8

Semana 3

Para cada diagrama:

Algo que debe ser cierto
Algo que podría ser cierto o falso
Algo que no puede ser cierto

¿Qué crees que pasará con cada percha si se agrega otro cuadrado a cada lado? ¿Qué pasa con un círculo con peso x?
Si 1 triángulo verde pesa 12.6 gramos, ¿cuál es el peso de 1 cuadrado azul?
Podemos usar la ecuación y = 3x para describir la percha de la derecha. ¿Cómo sería la percha para x + 3y = 2x + y?
Si y = 3x, ¿está equilibrada la percha para x + 3y = 2x + y? Explica o muestra cómo lo sabes.

ES

www.illustrativemathematics.org CC-BY-NC

Instructions:

Everyone try question A—give everyone 2-4 minutes of quiet think time and then start talking about your ideas.
Sentence/question starters for grown-ups and kids:

Tell me more about . . .
Are there more ways we can try?
Can you help me understand . . . ?
How do you see it? Can you show me with a drawing?

Keep going with any other problems that feel familiar and do-able.
Are you ready for more?

Reflect on what you did and how it went.
Connect with a classmate and ask them about their ideas.

Highlight below to see solutions:

A. Sample response:

Left1. The triangle is heavier than the square.

Left2. The triangle could weigh 10 ounces and the square could weigh 6 ounces.

Left3. The square and the triangle weigh the same.

Right1. One triangle weighs the same as three squares.

Right2. The triangle weighs three pounds and each square weighs one pound.

Right3. One square is heavier than the triangle.

B. Sample response: I think the hangers will stay the same since equal weight is being added to each side.

C. Responses vary.

D. 1 square and 2 triangles on the left side of the hanger, 2 squares and 1 triangle on the right side.

E. No, the hanger is not balance. Sample response: The hanger for x + 3y = 2x + y has the weight of 10 x’s on the left but only 5 on the right, so it is not in balance.

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Week 4

Talking Math 6–8

Launch

What do you notice?

What do you wonder?

A recipe calls for 6 cups of water and 2 scoops of lemonade powder.

www.illustrativemathematics.org CC-BY-NC

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Week 4

Talking Math 6–8

Write a rule that tells how many cups, y, of water would be needed for x scoops of lemonade powder for Han’s recipe.
Sketch a graph of the number of cups of scoops of lemonade powder vs. cups of water needed for Han’s recipe. Would the point (x, y) = (6, 15) be on the graph? What does that tell you about Clare’s recipe?
What is the slope of the graph? How is it related to the rule from question C?

Han is making a lot of lemonade. What are some amounts of water and lemonade powder he could use to make more lemonade that tastes the same?
Clare makes a recipe with 15 cups of water and 6 scoops of lemonade powder. Will her recipe taste the same as Han’s?

A recipe calls for 6 cups of water and 2 scoops of lemonade powder.

www.illustrativemathematics.org CC-BY-NC

Instructions:

Everyone try question A—give everyone 2-4 minutes of quiet think time and then start talking about your ideas.
Sentence/question starters for grown-ups and kids:

Tell me more about . . .
Are there more ways we can try?
Can you help me understand . . . ?
How do you see it? Can you show me with a drawing?

Keep going with any other problems that feel familiar and do-able.
Are you ready for more?

Reflect on what you did and how it went.
Connect with a classmate and ask them about their ideas.

Highlight below to see solutions:

A. Sample responses: 3 cups water, 2 scoops powder. 1 cup water, ⅓ scoop powder.

B. Sample response: It will not. If Han uses 15 cups of water, he needs 5 scoops of powder instead of 6 scoops.

C. Sample response: y = 3x

D. Sample response: Graphs vary depending on what scale is chosen for the axes, but should all look like either a set of points where the y-value is always 3 times the x-value or a line that starts at (0,0) and extends through the point (2, 6). The point (6, 15) will not be on the graph for Han’s recipe because for every 1 scoop of powder, Han needs 3 cups of water. If Han uses 6 scoops, he would need 18 cups of water. This tells me that Clare’s recipe is more lemon-y since she uses less water for the same amount of powder.

E. Sample response: The slope of the graph is 3. This is the same as the coefficient of x in the equation y = 3x for question C.

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Semana 4

Talking Math 6–8

Introducción

¿Qué observas?

¿Qué te preguntas?

Una receta de limonada dice: "Mezcle 2 cucharadas de polvo de limonada con 6 tazas de agua"

ES

www.illustrativemathematics.org CC-BY-NC

20 of 40

Talking Math 6–8

Semana 4

Escribe una regla que indique cuántas tazas, y, de agua serían necesarias para x cucharadas de limonada en polvo para la receta de Han.
Dibuje un gráfico del número de tazas de cucharadas de polvo de limonada versus tazas de agua necesarias para la receta de Han. ¿El punto (x, y) = (6, 15) estaría en la gráfica? ¿Qué te dice eso sobre la receta de Clare?
¿Cuál es la pendiente de la gráfica? ¿Cómo se relaciona con la regla de la pregunta C?

Han está haciendo mucha limonada. ¿Cuáles son algunas cantidades de agua y polvo de limonada que podría usar para hacer más limonada que tenga el mismo sabor?
Clare hace una receta con 15 tazas de agua y 6 cucharadas de limonada en polvo. ¿Su receta tendrá el mismo sabor que la de Han?

Una receta de limonada dice: "Mezcle 2 cucharadas de polvo de limonada con 6 tazas de agua"

ES

www.illustrativemathematics.org CC-BY-NC

Instructions:

Everyone try question A—give everyone 2-4 minutes of quiet think time and then start talking about your ideas.
Sentence/question starters for grown-ups and kids:

Tell me more about . . .
Are there more ways we can try?
Can you help me understand . . . ?
How do you see it? Can you show me with a drawing?

Keep going with any other problems that feel familiar and do-able.
Are you ready for more?

Reflect on what you did and how it went.
Connect with a classmate and ask them about their ideas.

Highlight below to see solutions:

A. Sample responses: 3 cups water, 2 scoops powder. 1 cup water, ⅓ scoop powder.

B. Sample response: It will not. If Han uses 15 cups of water, he needs 5 scoops of powder instead of 6 scoops.

C. Sample response: y = 3x

D. Sample response: Graphs vary depending on what scale is chosen for the axes, but should all look like either a set of points where the y-value is always 3 times the x-value or a line that starts at (0,0) and extends through the point (2, 6). The point (6, 15) will not be on the graph for Han’s recipe because for every 1 scoop of powder, Han needs 3 cups of water. If Han uses 6 scoops, he would need 18 cups of water. This tells me that Clare’s recipe is more lemon-y since she uses less water for the same amount of powder.

E. Sample response: The slope of the graph is 3. This is the same as the coefficient of x in the equation y = 3x for question C.

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Week 5

Talking Math 6–8

Launch

What is the same about their workouts?

What is different about their workouts?

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22 of 40

Week 5

Talking Math 6–8

Who was running faster? How do you know?
Choose one runner and complete these sentences: “For every minute of running, the runner went __________ miles. For every mile of running, the runner took __________ minutes.”
Noah is running outside, trying to keep up a steady pace of 6 miles per hour. He remembers that at noon, he was passing the half-mile marker. If he passes the 2-mile marker at 12:15, is he meeting his goal of 6 miles per hour?

www.illustrativemathematics.org CC-BY-NC

Instructions:

Everyone try question A—give everyone 2-4 minutes of quiet think time and then start talking about your ideas.
Sentence/question starters for grown-ups and kids:

Tell me more about . . .
Are there more ways we can try?
Can you help me understand . . . ?
How do you see it? Can you show me with a drawing?

Keep going with any other problems that feel familiar and do-able.
Are you ready for more?

Reflect on what you did and how it went.
Connect with a classmate and ask them about their ideas.

Highlight below to see solutions:

A. Sample responses: The first runner had a pace of 8 minutes per mile, so they were faster than the second runner whose pace of 10 minutes per mile means they take an extra 2 minutes to run the same distance.

B. Sample responses: For every minute of running, the runner went 1/8 miles. For every mile of running, the runner took 8 minutes. Or, For every minute of running, the runner went 1/10 miles. For every mile of running, the runner took 10 minutes.

C. Sample response: If Noah runs 1 mile in 15 minutes (noon to 12:15), then at that pace he will run 4 miles in an hour and not meet his goal.

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Talking Math 6–8

Semana 5

Introducción

¿Qué tienen en común los dos entrenamientos? ��¿En qué se diferencian los dos entrenamientos?

ES

www.illustrativemathematics.org CC-BY-NC

24 of 40

Talking Math 6–8

Semana 5

¿quién estaba corriendo más rápido? Explica tu razonamiento.
Elija un corredor y complete estas oraciones: “Por cada minuto de carrera, el corredor corrió _________ millas. Por cada milla de carrera, el corredor tardó _________ minutos ".
Noah está corriendo afuera, tratando de mantener un ritmo constante de 6 millas por hora. Él recuerda que al mediodía, estaba pasando el marcador de media milla. Si pasa el marcador de 2 millas a las 12:15, ¿está cumpliendo su objetivo de 6 millas por hora?

ES

www.illustrativemathematics.org CC-BY-NC

Instructions:

Everyone try question A—give everyone 2-4 minutes of quiet think time and then start talking about your ideas.
Sentence/question starters for grown-ups and kids:

Tell me more about . . .
Are there more ways we can try?
Can you help me understand . . . ?
How do you see it? Can you show me with a drawing?

Keep going with any other problems that feel familiar and do-able.
Are you ready for more?

Reflect on what you did and how it went.
Connect with a classmate and ask them about their ideas.

Highlight below to see solutions:

A. Sample responses: The first runner had a pace of 8 minutes per mile, so they were faster than the second runner whose pace of 10 minutes per mile means they take an extra 2 minutes to run the same distance.

B. Sample responses: For every minute of running, the runner went 1/8 miles. For every mile of running, the runner took 8 minutes. Or, For every minute of running, the runner went 1/10 miles. For every mile of running, the runner took 10 minutes.

C. Sample response: If Noah runs 1 mile in 15 minutes (noon to 12:15), then at that pace he will run 4 miles in an hour and not meet his goal.

25 of 40

Week 6

Talking Math 6–8

Launch

I have $2 in my pocket. What might be in my pocket?

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26 of 40

Week 6

Talking Math 6–8

Jada told Noah that she has $2 worth of quarters and dimes in her pocket and 17 coins all together.

What are some things that cannot be true about the coins in Jada’s pocket. Explain how you know.
Here is a table that shows some combinations of quarters and dimes that are worth $2. Complete the table.
Is it possible for Jada to have 4 quarters and 13 dimes in her pocket? Explain how you know.
How many quarters and dimes must Jada have? Explain your reasoning.

number of quarters	number of dimes
0	20
4
	0
	5

www.illustrativemathematics.org CC-BY-NC

Instructions:

Everyone try question A—give everyone 2-4 minutes of quiet think time and then start talking about your ideas.
Sentence/question starters for grown-ups and kids:

Tell me more about . . .
Are there more ways we can try?
Can you help me understand . . . ?
How do you see it? Can you show me with a drawing?

Keep going with any other problems that feel familiar and do-able.
Are you ready for more?

Contact a classmate and ask them about their ideas.
Reflect on what you did and how it went.

Highlight below to see solutions:

A. Sample response: Jada cannot have 20 dimes and 0 quarters because that is not 17 coins.

B. (0, 20); (4, 10); (8, 0); (6, 5); (2, 15)

C. No. Even though 4 quarters and 13 dimes is 17 coins, they are not worth $2. (They are worth $2.30.)

D. 2 quarters and 15 dimes. Sample explanation: This is 17 coins that are worth $2 because 2*0.25 + 15*0.10 = 2.00. It is the only combination of quarters and dimes that appears both in the table and as a point on the graph.

27 of 40

Semana 6

Talking Math 6–8

Introducción

Tengo $ 2 en mi bolsillo. ¿Qué podría haber en mi bolsillo?

ES

www.illustrativemathematics.org CC-BY-NC

28 of 40

Talking Math 6–8

Semana 6

ES

Jada le dijo a Noah que ella tiene en su bolsillo $2 en monedas de veinticinco centavos y de diez centavos, y que en total tiene 17 monedas.

¿Cuáles son algunas cosas que no pueden ser ciertas acerca de las monedas en el bolsillo de Jada? Explica tu razonamiento.
Esta es una tabla que muestra algunas combinaciones de monedas de veinticinco centavos y de diez centavos que valen $2. Completa la tabla.
¿Es posible que Jada tenga 4 monedas de veinticinco centavos y 13 de diez centavos en su bolsillo? Explica tu razonamiento.
¿Cuántas monedas de veinticinco centavos y de diez centavos debe tener Jada? Explica tu razonamiento.

número de monedas de veinticinco centavos	número de monedas de diez centavos
0	20
4
	0
	5

www.illustrativemathematics.org CC-BY-NC

Instructions:

Everyone try question A—give everyone 2-4 minutes of quiet think time and then start talking about your ideas.
Sentence/question starters for grown-ups and kids:

Tell me more about . . .
Are there more ways we can try?
Can you help me understand . . . ?
How do you see it? Can you show me with a drawing?

Keep going with any other problems that feel familiar and do-able.
Are you ready for more?

Reflect on what you did and how it went.
Connect with a classmate and ask them about their ideas.

Highlight below to see solutions:

A. Sample response: Jada cannot have 20 dimes and 0 quarters because that is not 17 coins.

B. (0, 20); (4, 10); (8, 0); (6, 5); (2, 15)

C. No. Even though 4 quarters and 13 dimes is 17 coins, they are not worth $2. (They are worth $2.30.)

D. 2 quarters and 15 dimes. Sample explanation: This is 17 coins that are worth $2 because 2*0.25 + 15*0.10 = 2.00. It is the only combination of quarters and dimes that appears both in the table and as a point on the graph.

29 of 40

Week 7

Talking Math 6–8

Launch

How are these images alike?

How are they different?

www.illustrativemathematics.org CC-BY-NC

30 of 40

Week 7

Talking Math 6–8

Elena says that 2 green triangles cover the same area as 1 blue rhombus. What other sets of shapes cover the same area?
Pick one of the patterns. Which shape covers more of the plane: blue rhombuses, red trapezoids, or green triangles? Explain how you know.
If the area covered by a pattern is 24 square units, how much area is covered by green triangles?
Is there a point on either of the patterns so that you could rotate the pattern about the point 180 degrees, and get the same image? How can you tell?

www.illustrativemathematics.org CC-BY-NC

Instructions:

Everyone try question A—give everyone 2-4 minutes of quiet think time and then start talking about your ideas.
Sentence/question starters for grown-ups and kids:

Tell me more about . . .
Are there more ways we can try?
Can you help me understand . . . ?
How do you see it? Can you show me with a drawing?

Keep going with any other problems that feel familiar and do-able.
Are you ready for more?

Contact a classmate and ask them about their ideas.
Reflect on what you did and how it went.

Highlight below to see solutions:

A. Sample responses: 10 green triangles and 5 rhombus; 9 green triangles and 3 red trapezoid; 4 red trapezoids and 6 blue rhombuses.

B. In both Patterns 1 and 2, more of the plane is covered by red trapezoids than green triangles or blue rhombuses. Explanations vary.

C. Each green triangle is ⅛ square unit and each pattern has a total of 56 green triangles, so the total area covered is ⅛ * 56 = 7 square units.

D. No. Sample response: The overall shape would be the same if you picked the point where the middle 4 hexagons touch and rotate the pattern 180 degrees, but the individual patterns would not match up. For example, the top of the first pattern has part of a red trapezoid, two triangles, and part of another red trapezoid. After being rotated, the top would be a green triangle, a red trapezoid, and a green triangle.

31 of 40

Semana 7

Talking Math 6–8

Introducción

¿En qué se parecen estas imágenes?

¿En qué se diferencian?

ES

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32 of 40

Talking Math 6–8

Semana 7

ES

Elena dice que 2 triángulos verdes cubren la misma área que 1 rombo azul. ¿Qué otros groupos de formas cubren la misma área?
Elije uno de los modelos. ¿Qué figuras cubren una mayor parte del plano: ¿rombos azules, trapecios rojos o triángulos verdes? Explica cómo lo sabes.
Si el área cubierta por un modelo es de 24 unidades cuadradas, ¿cuánta área está cubierta por triángulos verdes?
¿Hay algún punto en cualquiera de los modelos que te permita rotar el modelo alrededor del punto 180 grados y obtener la misma imagen? ¿Cómo puedes saberlo?

www.illustrativemathematics.org CC-BY-NC

Instructions:

Everyone try question A—give everyone 2-4 minutes of quiet think time and then start talking about your ideas.
Sentence/question starters for grown-ups and kids:

Tell me more about . . .
Are there more ways we can try?
Can you help me understand . . . ?
How do you see it? Can you show me with a drawing?

Keep going with any other problems that feel familiar and do-able.
Are you ready for more?

Contact a classmate and ask them about their ideas.
Reflect on what you did and how it went.

Highlight below to see solutions:

A. Sample responses: 10 green triangles and 5 rhombus; 9 green triangles and 3 red trapezoid; 4 red trapezoids and 6 blue rhombuses.

B. In both Patterns 1 and 2, more of the plane is covered by red trapezoids than green triangles or blue rhombuses. Explanations vary.

C. Each green triangle is ⅛ square unit and each pattern has a total of 56 green triangles, so the total area covered is ⅛ * 56 = 7 square units.

D. No. Sample response: The overall shape would be the same if you picked the point where the middle 4 hexagons touch and rotate the pattern 180 degrees, but the individual patterns would not match up. For example, the top of the first pattern has part of a red trapezoid, two triangles, and part of another red trapezoid. After being rotated, the top would be a green triangle, a red trapezoid, and a green triangle.

33 of 40

Week 8

Talking Math 6–8

Launch

Which one doesn’t belong?

www.illustrativemathematics.org CC-BY-NC

34 of 40

Week 8

Talking Math 6–8

The side length of each square on the grid is 1 unit.

What types of triangles do you see?
Which triangle has the greatest area? Show your reasoning.
Which triangle has the longest side length? Explain how you know.

www.illustrativemathematics.org CC-BY-NC

Instructions:

Everyone try question A—give everyone 2-4 minutes of quiet think time and then start talking about your ideas.
Sentence/question starters for grown-ups and kids:

Tell me more about . . .
Are there more ways we can try?
Can you help me understand . . . ?
How do you see it? Can you show me with a drawing?

Keep going with any other problems that feel familiar and do-able.
Are you ready for more?

Contact a classmate and ask them about their ideas.
Reflect on what you did and how it went.

Highlight below to see solutions:

A. S is a right triangle, T and V are obtuse triangles, U is an acute triangle.

B. S has area 6 u²; T has area 14 u²; U has area 12 u²; V has area 12 u².

C. Triangles T and V each have a side with length about 8.94 units. Sample explanations: I know this by drawing a right triangle whose hypotenuse is the longest side of T (or V) and legs 4 units and 8 units and then using the Pythagorean Theorem. Or, I used tracing paper to trace the longest side of triangle T and found it was the same length as the longest side of triangle V and longer than all the other sides. Lining up the traced line along the grid showed it was a little less than 9 units long.

35 of 40

Semana 8

Talking Math 6–8

Introducción

¿Cuál es diferente?

ES

www.illustrativemathematics.org CC-BY-NC

36 of 40

Talking Math 6–8

Semana 8

ES

La longitud del lado de cada cuadrado de la cuadrícula es 1 unidad.

¿Qué tipos de triángulos ves?
¿Qué triángulo tiene el área más grande? Muestra tu razonamiento.
¿Qué triángulo tiene la longitud lateral más larga? Explica tu razonamiento.

www.illustrativemathematics.org CC-BY-NC

Instructions:

Everyone try question A—give everyone 2-4 minutes of quiet think time and then start talking about your ideas.
Sentence/question starters for grown-ups and kids:

Tell me more about . . .
Are there more ways we can try?
Can you help me understand . . . ?
How do you see it? Can you show me with a drawing?

Keep going with any other problems that feel familiar and do-able.
Are you ready for more?

Contact a classmate and ask them about their ideas.
Reflect on what you did and how it went.

Highlight below to see solutions:

A. S is a right triangle, T and V are obtuse triangles, U is an acute triangle.

B. S has area 6 u²; T has area 14 u²; U has area 12 u²; V has area 12 u².

C. Triangles T and V each have a side with length about 8.94 units. Sample explanations: I know this by drawing a right triangle whose hypotenuse is the longest side of T (or V) and legs 4 units and 8 units and then using the Pythagorean Theorem. Or, I used tracing paper to trace the longest side of triangle T and found it was the same length as the longest side of triangle V and longer than all the other sides. Lining up the traced line along the grid showed it was a little less than 9 units long.

37 of 40

Week 9

Talking Math 6–8

Launch

What do you notice?

What do you wonder?

www.illustrativemathematics.org CC-BY-NC

38 of 40

Week 9

Talking Math 6–8

A ladybug and an ant move at constant speeds. Lines u and v show how much time it took them to travel different distances.

The ant moves faster than the ladybug. Which line shows the ladybug’s movement? Explain your reasoning.
How long does it take the ladybug to travel 12 cm?
Imagine a bug that is moving twice as fast as the ladybug. Plot this bug’s time for different distances on the coordinate axes, and connect them with a line.
Write an equation for each of the three lines.

www.illustrativemathematics.org CC-BY-NC

Instructions:

Everyone try question A—give everyone 2-4 minutes of quiet think time and then start talking about your ideas.
Sentence/question starters for grown-ups and kids:

Tell me more about . . .
Are there more ways we can try?
Can you help me understand . . . ?
How do you see it? Can you show me with a drawing?

Keep going with any other problems that feel familiar and do-able.
Are you ready for more?

Contact a classmate and ask them about their ideas.
Reflect on what you did and how it went.

Highlight below to see solutions:

A. Sample reasoning: Line u shows a bug going 2 cm in 1 second while line v shows a bug going 3 cm in 1 second. The ladybug must be line v since that bug went further in the same amount of time.

B. 4 seconds

C. A line starting at (0, 0) that extends through the point (4, 1) and (12, 3).

D. Possible response: Equations are ladybug: y = ½ x, ant: y = ⅓ x, new bug: y = ¼ x (twice as fast as ladybug), where x represents the distance traveled and y represents elapsed time.

39 of 40

Semana 9

Talking Math 6–8

Introducción

ES

Introducción

¿Qué observas?

¿Qué te preguntas?

www.illustrativemathematics.org CC-BY-NC

40 of 40

Talking Math 6–8

Semana 9

ES

Un escarabajo y una hormiga se mueven a rapidez constante. Las rectas u y v muestran cuánto tiempo les llevó recorrer diferentes distancias.

La hormiga se mueve más rápido que la mariquita. ¿Qué recta muestra el movimiento del escarabajo? Explica tu razonamiento.
¿Cuánto tarda el escarabajo en desplazarse 12 cm?
Imagina un insecto que se mueve el doble de rápido que el escarabajo. Trace el tiempo de este insecto para diferentes distancias en los ejes de coordenadas y conéctelos con una línea.
Escribe una ecuación para cada una de las tres rectas.

www.illustrativemathematics.org CC-BY-NC

Instructions:

Everyone try question A—give everyone 2-4 minutes of quiet think time and then start talking about your ideas.
Sentence/question starters for grown-ups and kids:

Tell me more about . . .
Are there more ways we can try?
Can you help me understand . . . ?
How do you see it? Can you show me with a drawing?

Keep going with any other problems that feel familiar and do-able.
Are you ready for more?

Reflect on what you did and how it went.
Connect with a classmate and ask them about their ideas.

Highlight below to see solutions:

A. Sample reasoning: Line u shows a bug going 2 cm in 1 second while line v shows a bug going 3 cm in 1 second. The ladybug must be line v since that bug went further in the same amount of time.

B. 4 seconds

C. A line starting at (0, 0) that extends through the point (4, 1) and (12, 3).

D. Possible response: Equations are ladybug: y = ½ x, ant: y = ⅓ x, new bug: y = ¼ x (twice as fast as ladybug), where x represents the distance traveled and y represents elapsed time.