Unit 4, Lesson 5
How Many Groups? (Part 2)
Grade 6
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In this Warm-up, students continue to think of division in terms of equal-size groups, using fraction strips as an additional tool for reasoning.
Monitor for how students transition from concrete questions (the first three) to symbolic ones (the last three). Framing division expressions as “How many of this fraction is in that number?” may not yet be intuitive to students. They will further explore that connection in a later activity. For now, support them using whole-number examples such as by asking “How do you interpret
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, such as by first writing an equivalent fraction.
As students work, identify those who are able to modify their reasoning effectively, even if the approach may not be efficient (such as adding a row of
Give students 2–3 minutes of quiet work time.
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.
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.
Because the fraction strips do not show tenths, students might be unsure how to approach the last question. Ask questions such as:
Focus the discussion on how students interpreted division expressions such as
For the last question, if no students mention using a unit fraction that is equivalent to
MLR8: Discussion Supports
This activity serves two purposes: to explicitly bridge “How many of this in that?” questions to division expressions, and to explore division situations in which the quotients are not 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 (
As students discuss in groups, listen for their explanations for the question “How many rhombuses are in a trapezoid?” Select two students to share later—one person to elaborate on Diego’s argument, and another to support Jada’s argument.
This activity works best when each student has access to pattern blocks. If pattern blocks are not available, consider using the digital version of the activity. In the digital version, students use an applet to investigate fractional relationships with pattern blocks. The applet allows students to place and compare pattern blocks with or without a grid.
Focus the whole-class discussion on the last two questions. In particular, discuss how the visual representation helps us reason about Jada’s and Diego’s points of view, as well as the connections between the verbal and numerical representations of the situation.
Select the two previously identified students to explain why Diego or Jada is correct. Display a visual representation of “How many rhombuses are in a trapezoid?” for all to see (as shown here), or use the applet for illustration.
To highlight number of groups and size of one group in the problem, discuss questions such as:
If students struggle to compare the remainder to a rhombus, ask: “How many triangles are in a trapezoid?” and point out that the answer is “3 triangles.” Here, the answer to “how many rhombuses are in a trapezoid?” would be “(some number of) rhombuses.”
The fact that there are two 1 wholes to keep track of may be a source of confusion (the trapezoid represents the quantity 1 and the rhombus represents 1 group). Students will have opportunities to make clearer distinctions between the two 1 wholes in upcoming activities.
Suppose one batch of cookies requires
We can think of the question as being: “How many
Let’s use pattern blocks to visualize the situation and say that a hexagon is 1 whole.
Diagram of 4 hexagons. Each hexagon is made up of 3 rhombuses using pattern blocks. 6 pairs of rhombuses are shown by pairs of arrows and labeled 1, 2, 3, 4, 5, and 6.
Since 3 rhombuses make a hexagon, 1 rhombus represents
We can see that 6 pairs of rhombuses make 4 hexagons, so there are 6 groups of
Other kinds of diagrams can also help us reason about equal-size groups involving fractions. This example shows how we might reason about the same question asked earlier: “How many
Diagram with 4 rectangles, partitioned into thirds. In the first rectangle two parts are each labeled one third. Across all 4 rectangles, 6 pairs of one third pieces are circled.
We can see each “cup” partitioned into thirds, and that there are 6 groups of
Help us improve by sharing suggestions or reporting issues.
Tell students that they will use pattern blocks again to think about equal-size groups involving fractions, but that this time the trapezoid represents 1 whole.
Arrange students in groups of 3–4. Provide access to pattern blocks and geometry toolkits. Give students 10 minutes of quiet work time for the first three questions and a few minutes to discuss their responses and collaborate on the last question.
Your teacher will give you pattern blocks. Use them to answer the questions.
If the trapezoid represents 1 whole, what does each of the other shapes represent? Be prepared to show or explain your reasoning.
1 triangle
1 rhombus
1 hexagon
Use pattern blocks to represent each multiplication equation. Use the trapezoid to represent 1 whole. Sketch or trace the blocks to record your representation.
Diego and Jada were asked “How many rhombuses are in a trapezoid?”
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?”
For each situation:
When writing equations, students may reverse the values for the divisor and dividend. Encourage students to think about the meanings of the quantities in context. Consider asking:
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