In this warm-up, students continue to think of division in terms of equal-sized groups, using fraction strips as an additional tool for reasoning.

You are watching: How many rhombuses make a hexagon

Notice how students transition from concrete questions (the first three) to symbolic ones (the last three). Framing division expressions as “how many of this fraction in that number?” may not yet be intuitive to students. They will further explore that connection in this lesson. For now, support them using whole-number examples (e.g., ask: “how do you interpret (6 div 2)?”).

The divisors used here involve both unit fractions and non-unit fractions. The last question shows a fractional divisor that is not on the fraction strips. This encourages students to transfer the reasoning used with fraction strips to a new problem, or to use an additional strategy (e.g., by first writing an equivalent fraction).

As students work, notice those who are able to modify their reasoning effectively, even if the approach may not be efficient (e.g., adding a row of (frac 110)s to the fraction strips). Ask them to share later.

### Launch

Give students 2–3 minutes of quiet work tmslsec.come.

### Student Facing

Write a fraction or whole number as an answer for each question. If you get stuck, use the fraction strips. Be prepared to share your reasoning.

How many (frac 12)s are in 2?How many (frac 15)s are in 3?How many (frac 18)s are in (1frac 14)?(1 div frac 26 = ?)(2 div frac 29 = ?)(4 div frac 210 = ?)

Description: Fraction strips depicting 2 in 8 different ways, by rows. First row, two 1s. Second row, 4 of the fraction one over two. Third row, 6 of the fraction one over three. Fourth row, 8 of the fraction one over four. Fifth row, 10 of the fraction one over five. Sixth row, 12 of the fraction one over six. Seventh row, 16 of the fraction one over eight. Eight row, 18 of the fraction one over nine.

Since the fraction strips do not show tenths, students might think that it is mslsec.compossible to answer the last question. Ask them if they can think of another fraction that is equivalent to (frac210).

For each of the first five questions, select a student to share their response and ask the class to indicate whether they agree or disagree.

Focus the discussion on two things: how students interpreted expressions such as (1 div frac26), and on how they reasoned about (4 div frac 210). Select a few students to share their reasoning.

For the last question, highlight strategies that are effective and efficient, such as using a unit fraction that is equivalent to (frac 210), finding out how many groups of (frac15) are in 1 and then multiplying it by 4, etc.

## 5.2: More Reasoning with Pattern Blocks (25 minutes)

CCSS Standards

Routines and Materials

Instructional Routines

Required Materials

### Activity

This activity serves two purposes: to explicitly bridge “how many of this in that?” questions and division expressions, and to explore division situations in which the quotients are not whole numbers. (Students explored smslsec.comilar questions previously, but the quotients were whole numbers.)

Once again students move from reasoning concretely and visually to reasoning symbolically. They start by thinking about “how many rhombuses are in a trapezoid?” and then express that question as multiplication((? oldcdot frac23 = 1) or (frac 23 oldcdot ,? = 1)) and division ((1 div frac23)). Students think about how to deal with a remainder in such problems.

As students discuss in groups, listen for their explanations for the question “How many rhombuses are in a trapezoid?” Select a couple of students to share later—one person to elaborate on Diego"s argument, and another to support Jada"s argument.

Arrange students in groups of 3–4. Provide access to pattern blocks and geometry toolkits. Give students 10 minutes of quiet work tmslsec.come for the first three questions and a few minutes to discuss their responses and collaborate on the last question.

Classrooms with no access to pattern blocks or those using the digital materials can use the provided applet. Physical pattern blocks are still preferred, however.

Representation: Develop Language and Symbols. Display or provide charts with symbols and meanings. Emphasize the difference between this activity where students must find what fraction of a trapezoid each of the shapes represents, compared to the hexagon in the previous lesson. Create a display that includes an mslsec.comage of each shape labeled with the name and the fraction it represents of a trapezoid. Keep this display visible as students move on to the next problems.Supports accessibility for: Conceptual processing; Memory
Use the pattern blocks in the applet to answer the questions. (If you need help aligning the pieces, you can turn on the grid.)

If the trapezoid represents 1 whole, what do each of these other shapes represent? Be prepared to explain or show your reasoning.

1 triangle

1 rhombus

1 hexagon

Use pattern blocks to represent each multiplication equation. Use the trapezoid to represent 1 whole.

(3 oldcdot frac 13=1)

(3 oldcdot frac 23=2)

Diego and Jada were asked “How many rhombuses are in a trapezoid?”

Diego says, “(1frac 13). If I put 1 rhombus on a trapezoid, the leftover shape is a triangle, which is (frac 13) of the trapezoid.”Jada says, “I think it’s (1frac12). Since we want to find out ‘how many rhombuses,’ we should compare the leftover triangle to a rhombus. A triangle is (frac12) of a rhombus.”

Do you agree with either of them? Explain or show your reasoning.

Select all the equations that can be used to answer the question: “How many rhombuses are in a trapezoid?”

(frac 23 div ? = 1)

(? oldcdot frac 23 = 1)

(1 div frac 23 = ?)

(1 oldcdot frac 23 = ?)

(? div frac 23 = 1)

Arrange students in groups of 3–4. Provide access to pattern blocks and geometry toolkits. Give students 10 minutes of quiet work tmslsec.come for the first three questions and a few minutes to discuss their responses and collaborate on the last question.

Classrooms with no access to pattern blocks or those using the digital materials can use the provided applet. Physical pattern blocks are still preferred, however.

Representation: Develop Language and Symbols.Display or provide charts with symbols and meanings. Emphasize the difference between this activity where students must find what fraction of a trapezoid each of the shapes represents, compared to the hexagon in the previous lesson. Create a display that includes an mslsec.comage of each shape labeled with the name and the fraction it represents of a trapezoid. Keep this display visible as students move on to the next problems.Supports accessibility for: Conceptual processing; Memory
Your teacher will give you pattern blocks. Use them to answer the questions.

If the trapezoid represents 1 whole, what do each of the other shapes represent? Be prepared to show or explain your reasoning.

Use pattern blocks to represent each multiplication equation. Use the trapezoid to represent 1 whole.

See more: What Factors Do 16 And 24 Have In Common Factor Of 16 And 24 Gcf(16,24)

(3 oldcdot frac 13=1)(3 oldcdot frac 23=2)

Diego and Jada were asked “How many rhombuses are in a trapezoid?”

Diego says, “(1frac 13). If I put 1 rhombus on a trapezoid, the leftover shape is a triangle, which is (frac 13) of the trapezoid.”Jada says, “I think it’s (1frac12). Since we want to find out ‘how many rhombuses,’ we should compare the leftover triangle to a rhombus. A triangle is (frac12) of a rhombus.”

Do you agree with either of them? Explain or show your reasoning.

Select all the equations that can be used to answer the question: “How many rhombuses are in a trapezoid?”