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

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Notice just how students transition from concrete questions (the first three) come symbolic persons (the last three). Framing division expressions as “how many of this fraction in the number?” might not yet be intuitive come students. They will certainly further discover that link in this lesson. Because that now, support them making use of whole-number examples (e.g., ask: “how carry out you translate $$6 \div 2$$?”).

The divisors used below involve both unit fractions and non-unit fractions. The last concern shows a fractional divisor the is no on the fraction strips. This urges students to deliver the reasoning provided with fraction strips to a brand-new problem, or come use second strategy (e.g., by an initial writing an equivalent fraction).

As student work, an alert those who room able to change their thinking effectively, also if the approach may not be efficient (e.g., adding a row of $$\frac 110$$s to the portion strips). Questioning them to share later.

### Launch

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

### Student Facing

Write a fraction or totality number as solution for every question. If you acquire stuck, usage the portion strips. Be prepared to share her reasoning.

How many $$\frac 12$$s space in 2?How countless $$\frac 15$$s space in 3?How numerous $$\frac 18$$s room in $$1\frac 14$$?$$1 \div \frac 26 = ?$$$$2 \div \frac 29 = ?$$$$4 \div \frac 210 = ?$$

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

Since the fraction strips do not display tenths, students might think the it is historicsweetsballroom.compossible to answer the critical question. Ask them if they have the right to think of another portion that is indistinguishable to $$\frac210$$.

For every of the very first five questions, pick a college student to re-publishing their response and questioning the class to indicate whether castle agree or disagree.

Focus the conversation on 2 things: just how students interpreted expressions such as $$1 \div \frac26$$, and on exactly how they reasoned about $$4 \div \frac 210$$. Choose a few students come share their reasoning.

For the last question, highlight methods that are effective and also efficient, together as utilizing a unit portion that is tantamount to $$\frac 210$$, finding out how numerous groups of $$\frac15$$ room in 1 and also then multiply it by 4, etc.

## 5.2: an ext Reasoning with Pattern blocks (25 minutes)

CCSS Standards

Routines and also Materials

Instructional Routines

Required Materials

### Activity

This activity serves 2 purposes: to explicitly bridge “how plenty of of this in that?” concerns and department expressions, and also to explore division situations in i beg your pardon the quotients space not entirety numbers. (Students explored shistoricsweetsballroom.comilar questions previously, however the quotients were whole numbers.)

Once again students move from reasoning concretely and visually to reasoning symbolically. They start by thinking about “how plenty of rhombuses space in a trapezoid?” and also then express that inquiry as multiplication($$? \boldcdot \frac23 = 1$$ or $$\frac 23 \boldcdot \,? = 1$$) and division ($$1 \div \frac23$$). Students think about how to resolve a remainder in together problems.

As students talk about in groups, listen for their explanations because that the question “How numerous rhombuses room in a trapezoid?” choose a couple of students come share later—one person to intricate on Diego"s argument, and also another to support Jada"s argument.

Arrange students in groups of 3–4. Provide accessibility to sample blocks and geometry toolkits. Give students 10 minute of quiet work-related thistoricsweetsballroom.come for the first three questions and also a few minutes to talk about their responses and collaborate top top the last question.

Classrooms with no access to pattern block or those using the digital materials deserve to use the noted applet. Physical sample blocks room still preferred, however.

Representation: construct Language and also Symbols. Display or carry out charts v symbols and meanings. Emphasize the difference between this activity where students must uncover what portion of a trapezoid every of the forms represents, compared to the hexagon in the ahead lesson. Develop a screen that consists of an historicsweetsballroom.comage the each shape labeled v the name and the portion it represents of a trapezoid. Keep this display screen visible together students relocate on come the next problems.Supports availability for: theoretical processing; Memory
Use the pattern blocks in the applet come answer the questions. (If you need aid aligning the pieces, you deserve to turn top top the grid.)

If the trapezoid to represent 1 whole, what perform each of these other shapes represent? Be ready to explain or display your reasoning.

1 triangle

1 rhombus

1 hexagon

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

$$3 \boldcdot \frac 13=1$$

$$3 \boldcdot \frac 23=2$$

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

Diego says, “$$1\frac 13$$. If I put 1 rhombus top top a trapezoid, the leftover form is a triangle, i beg your pardon is $$\frac 13$$ that the trapezoid.”Jada says, “I think that $$1\frac12$$. Because we desire to find out ‘how plenty of rhombuses,’ we must compare the leftover triangle to a rhombus. A triangle is $$\frac12$$ the a rhombus.”

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

Select all the equations that deserve to be supplied to answer the question: “How countless rhombuses room in a trapezoid?”

$$\frac 23 \div ? = 1$$

$$? \boldcdot \frac 23 = 1$$

$$1 \div \frac 23 = ?$$

$$1 \boldcdot \frac 23 = ?$$

$$? \div \frac 23 = 1$$

Teachers through a valid work-related email deal with can click here to register or authorize in for totally free access to college student Response.

Arrange college student in teams of 3–4. Provide access to pattern blocks and geometry toolkits. Offer students 10 minute of quiet work-related thistoricsweetsballroom.come because that the very first three questions and a few minutes to comment on their responses and collaborate on the critical question.

Classrooms with no access to pattern block or those making use of the digital materials can use the noted applet. Physical sample blocks room still preferred, however.

Representation: develop Language and also Symbols.Display or administer charts with symbols and meanings. Emphasize the difference in between this activity where college student must uncover what portion of a trapezoid every of the shapes represents, contrasted to the hexagon in the previous lesson. Develop a display screen that includes an historicsweetsballroom.comage of each form labeled through the name and the portion it to represent of a trapezoid. Save this display visible together students relocate on come the following problems.Supports ease of access for: conceptual processing; Memory
Your teacher will provide you pattern blocks. Usage them come answer the questions.

If the trapezoid to represent 1 whole, what do each that the various other shapes represent? Be all set to present or define your reasoning.

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

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$$3 \boldcdot \frac 13=1$$$$3 \boldcdot \frac 23=2$$

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

Diego says, “$$1\frac 13$$. If I placed 1 rhombus ~ above a trapezoid, the leftover shape is a triangle, which is $$\frac 13$$ of the trapezoid.”Jada says, “I think that $$1\frac12$$. Because we desire to discover out ‘how numerous rhombuses,’ we should compare the leftover triangle to a rhombus. A triangle is $$\frac12$$ the a rhombus.”

Do you agree with either of them? describe or display your reasoning.

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