Lesson 1

The purpose of this Estimation Exploration is to invite students to think about small fractions of a quantity in context. The mosaic pictured here is made up of many small square tiles. They are arranged in a complex pattern and are not identical in size but students can relate the denominator of a fraction giving the size of each tile, relative to the whole mosaic, to the total number of tiles making the mosaic. This helps them think of a fraction with a large denominator which prepares them to think about the fraction \(\frac{1}{1,000}\) in this lesson. 

• Groups of 2
• Display the image.
• “¿Qué estimación sería muy alta?, ¿muy baja?, ¿razonable?” // “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time

• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

• “¿Cómo hicieron su estimación?” // “How did you make your estimate?” (I tried to estimate out how many tiles made the whole picture and then used that as my denominator.)
• “En varias de las siguientes lecciones vamos a examinar números muy pequeños y a descubrir cómo representarlos” // “We will be investigating very small numbers and how to represent them in the next several lessons.”

The purpose of this activity is for students to share what they know about one tenth and one hundredth, and consider what they might know about one thousandth. Students make a poster showing what they know about these numbers and then discuss different representations they made. If students show tenths, hundredths or thousandths on a number line or with base ten diagrams, highlight these representations in the synthesis, as they are familiar from grade 4.

This activity is meant to be an invitational opportunity for students to bring their lived experience into the math classroom. Consider taking a walk through the community where your students live and noticing places that decimals are seen and used. Take pictures or notes that capture the details of your observations. Be prepared to share these artifacts with students during the synthesis.

• Groups of 2
• Give students access to hundredths grids on the blackline master.
• Give students access to chart paper and colored pencils, crayons, or markers.
• “Con su compañero, hagan un póster que muestre lo que saben sobre estos números: 1 décimo, 1 centésimo y 1 milésimo” // “Work with your partner to make a poster showing what you know about the numbers 1 tenth, 1 hundredth, and 1 thousandth.”

• 6-8 minutes: partner work time
• Monitor for students who:
  • represent the numbers with fractions
  • represent the numbers with decimals
  • represent the numbers with diagrams

• Invite students to share their posters.
• “¿Cuáles son algunas maneras de representar 1 décimo?” // “What are some ways to represent 1 tenth?” (I can write it as a fraction or a decimal. I can divide a rectangle into 10 equal pieces. I can put it on the number line.)
• “¿Qué fue retador al representar 1 centésimo con un dibujo o con un diagrama?” // “What was challenging about representing 1 hundredth with a drawing or diagram?” (Answers vary. It was hard to divide something into 100 equal pieces because that’s a lot.)
• “¿Qué fue retador al representar 1 milésimo con un dibujo o con un diagrama?” // “What was challenging about representing 1 thousandth with a drawing or diagram?” (Dividing a rectangle into 1,000 pieces would take forever.)
• “En la siguiente actividad vamos a hacer y a comparar dibujos de estos números” // “In the next activity we will make and compare drawings of these numbers.”

The purpose of this activity is for students to represent tenths, hundredths, and thousandths with diagrams and decimals. The diagrams highlight the relationships between these quantities:

• there are 10 tenths in a whole
• there are 10 hundredths in a tenth
• there are 10 thousandths in a hundredth

This relationship can be seen when numbers are written as decimals or fractions. When students see this common relationship between the decimal place values they look for and make use of structure (MP7).

• Groups of 2
• “Hoy vamos a examinar algunos números realmente pequeños” // “Today we are going to investigate some really small numbers.”

• 2 minutes: independent work time
• 8 minutes: partner work time

If students do not explain how many small rectangular pieces are in the whole unit square, ask: “¿Cuántas partes pequeñas rectangulares habría en una fila del cuadrado unitario?” // “How many small rectangular pieces would be in one row of the unit square?”

• Display the last image.
• “¿Cuántos rectángulos pequeños sombreados hay en todo el cuadrado unitario? ¿Cómo lo saben?” // “How many of the tiny shaded rectangles are there in the whole unit square? How do you know?" (1,000 because there are 10 in the small square and 100 of those squares in the whole.)
• “De todo el cuadrado, ¿cuánto está sombreado?” // “How much of the whole square is shaded?” (There is one tiny rectangle shaded and there are 1,000 of those in the whole so \(\frac{1}{1,\!000}\) is shaded. There is \(\frac{1}{10}\) of \(\frac{1}{100}\) shaded. That’s \(\frac{1}{10} \times \frac{1}{100}\). There is \(\frac{1}{100}\div10\) because the hundredth is divided into 10 equal pieces.)
• “El número \(\frac{1}{1,\!000}\) también se puede escribir en forma decimal como 0.001. De forma similar a la fracción, al número decimal lo llamamos ‘milésima’” // “The number \(\frac{1}{1,\!000}\) can also be written in decimal form as 0.001. Like the fraction, we call it ‘thousandth.’”
• “¿Cómo piensan que se podría escribir \(\frac{4}{1,\!000}\) como un número decimal?” // “How do you think you would write \(\frac{4}{1,\!000}\) as a decimal?” (0.004)