In this warm-up, students continue to think of department in regards to equal-sized groups, using portion strips as second tool because that reasoning.

You are watching: How many rhombuses in a hexagon

Notice just how students shift from concrete inquiries (the very first three) come symbolic persons (the last three). Framing division expressions as “how plenty of of this portion in the number?” may not yet be intuitive come students. They will further discover that connection in this lesson. Because that now, assistance them using whole-number instances (e.g., ask: “how execute you translate (6 div 2)?”).

The divisors used below involve both unit fractions and non-unit fractions. The last concern shows a fractional divisor that is no on the fraction strips. This encourages students to transfer the reasoning used with fraction strips come a new problem, or to use secondary strategy (e.g., by first writing an tantamount fraction).

As students work, notice those who are able to modify their reasoning effectively, also if the method may not be effective (e.g., adding a heat of (frac 110)s come the fraction strips). Asking them to share later.

### Launch

Give student 2–3 minute of quiet work-related tmoment-g.come.

Student Facing

Write a fraction or whole number as response for each question. If you gain stuck, usage the fraction strips. Be ready to share your reasoning.How plenty of (frac 12)s are in 2?How many (frac 15)s space in 3?How many (frac 18)s room in (1frac 14)?(1 div frac 26 = ?)(2 div frac 29 = ?)(4 div frac 210 = ?)

**Description:**fraction strips showing 2 in 8 different ways, through rows. Very first row, two 1s. Second row, 4 that the portion one over two. 3rd row, 6 of the portion one end three. 4th row, 8 of the fraction one end four. 5th row, 10 the the portion one end five. 6th row, 12 that the portion one over six. Seventh row, 16 of the fraction one over eight. Eight row, 18 of the portion one over nine.

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Since the portion strips do not display tenths, students can think that it is moment-g.compossible to prize the critical question. Ask castle if they can think the another portion that is identical to (frac210).

For each of the first five questions, choose a student to share their response and questioning the course to indicate whether castle agree or disagree.

Focus the discussion on 2 things: how students understood expressions such together (1 div frac26), and also on exactly how they reasoned about (4 div frac 210). Choose a couple of students to share your reasoning.

For the critical question, highlight techniques that space effective and also efficient, such as utilizing a unit portion that is equivalent to (frac 210), finding the end how countless groups the (frac15) space in 1 and also then multiply it by 4, etc.

## 5.2: much more Reasoning v Pattern block (25 minutes)

CCSS Standards

Addressing

Routines and also Materials

Instructional Routines

Required Materials

### Activity

This activity serves two purposes: to explicitly bridge “how countless of this in that?” concerns and department expressions, and also to explore department situations in i beg your pardon the quotients room not totality numbers. (Students explored smoment-g.comilar questions previously, yet the quotients were totality numbers.)

Once again students move from thinking concretely and visually to thinking symbolically. They begin by thinking around “how many rhombuses space in a trapezoid?” and also then express that inquiry as multiplication((? oldcdot frac23 = 1) or (frac 23 oldcdot ,? = 1)) and division ((1 div frac23)). College student think about how to attend to a remainder in together problems.

As students comment on in groups, hear for your explanations because that the inquiry “How countless rhombuses are in a trapezoid?” pick a pair of students come share later—one human being to fancy on Diego"s argument, and also another to support Jada"s argument.

Arrange college student in groups of 3–4. Provide access to sample blocks and geometry toolkits. Provide students 10 minute of quiet occupational tmoment-g.come because that the an initial three questions and also a couple of minutes to comment on their responses and collaborate ~ above the last question.

Classrooms v no access to pattern block or those utilizing the digital materials deserve to use the detailed applet. Physical sample blocks are still preferred, however.

*Representation: build Language and also Symbols.*Display or carry out charts through symbols and also meanings. Emphasize the difference in between this task where student must find what fraction of a trapezoid each of the forms represents, contrasted to the hexagon in the previous lesson. Produce a display screen that includes an moment-g.comage of each form labeled with the name and the portion it to represent of a trapezoid. Store this display visible as students move on to the following problems.

*Supports availability for: conceptual processing; Memory*

Use the pattern blocks in the applet come answer the questions. (If you need help aligning the pieces, you deserve to turn ~ above the grid.)

If the trapezoid represents 1 whole, what perform each that these various other shapes represent? Be prepared to describe or present 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 also Jada were asked “How countless rhombuses space in a trapezoid?”

Diego says, “(1frac 13). If I put 1 rhombus top top a trapezoid, the leftover form is a triangle, i m sorry is (frac 13) of the trapezoid.”Jada says, “I think the (1frac12). Due to the fact that we want to discover out ‘how numerous rhombuses,’ we should compare the leftover triangle come a rhombus. A triangle is (frac12) of a rhombus.”Do you agree through either that them? describe or present your reasoning.

Select **all** the equations that can be provided to prize the question: “How countless rhombuses space in a trapezoid?”

(frac 23 div ? = 1)

(? oldcdot frac 23 = 1)

(1 div frac 23 = ?)

(1 oldcdot frac 23 = ?)

(? div frac 23 = 1)

Teachers through a valid work email resolve can click right here to register or authorize in for cost-free access to college student Response.

Arrange student in teams of 3–4. Provide accessibility to sample blocks and also geometry toolkits. Provide students 10 minute of quiet work-related tmoment-g.come because that the very first three questions and a few minutes to comment on their responses and collaborate ~ above the last question.

Classrooms with no accessibility to pattern block or those utilizing the digital materials can use the detailed applet. Physical pattern blocks room still preferred, however.

*Representation: build Language and also Symbols.*Display or administer charts through symbols and also meanings. Emphasize the difference between this activity where college student must discover what portion of a trapezoid each of the forms represents, contrasted to the hexagon in the previous lesson. Create a display that includes picture of each form labeled with the name and also the fraction it to represent of a trapezoid. Save this display screen visible together students move on to the following problems.

*Supports accessibility for: conceptual processing; Memory*

Your teacher will give you pattern blocks. Usage them come answer the questions.

If the trapezoid to represent 1 whole, what execute each of the various other shapes represent? Be prepared to present or describe your reasoning.

Use pattern block to stand for each multiplication equation. Usage the trapezoid to stand for 1 whole.

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

Diego and Jada were asked “How numerous rhombuses space in a trapezoid?”

Diego says, “(1frac 13). If I put 1 rhombus on a trapezoid, the leftover form is a triangle, i m sorry is (frac 13) the the trapezoid.”Jada says, “I think that (1frac12). Because we want to discover out ‘how many rhombuses,’ we have to compare the leftover triangle come a rhombus. A triangle is (frac12) the a rhombus.”Do friend agree v either that them? describe or present your reasoning.

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