standard_description: Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
problem_activity: 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 similar 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
(
\(? \boldcdot \frac23 = 1\)
or
\(\frac 23 \boldcdot \,? = 1\)
) 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.
Digital
Launch
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.
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 image 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
Student Facing
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.
\(3 \boldcdot \frac 13=1\)
\(3 \boldcdot \frac 23=2\)
Diego and Jada were asked “How many rhombuses are in a trapezoid?”
Diego says, “
\(1\frac 13\)
. 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
\(1\frac12\)
. 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?”
\(\frac 23  \div {?} = 1\)
\({?} \boldcdot \frac 23 = 1\)
\(1 \div \frac 23 = {?}\)
\(1 \boldcdot \frac 23 = {?}\)
\({?}  \div \frac 23 = 1\)
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
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.
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 image 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
Student Facing
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.
[IMG]
Use pattern blocks to represent each multiplication equation. Use the trapezoid to represent 1 whole.
\(3 \boldcdot \frac 13=1\)
\(3 \boldcdot \frac 23=2\)
Diego and Jada were asked “How many rhombuses are in a trapezoid?”
Diego says, “
\(1\frac 13\)
. 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
\(1\frac12\)
. 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?”
\(\frac 23 \div {?} = 1\)
\(? \boldcdot \frac 23 = 1\)
\(1 \div \frac 23 = {?}\)
\(1 \boldcdot \frac 23 = {?}\)
\({?} \div \frac23 = 1\)
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Some students may not notice that in this task, the trapezoid—not the hexagon—represents 1 whole. Encourage them to revisit the task statement to check.
Activity Synthesis
Focus the whole-class discussion on the last two questions, especially on how the visual representation helps us reason about Jada and Diego's points of view, and on the connections between the verbal and numerical representations of the situation.
Select two previously identified students to explain why Diego or Jada are correct. Display a visual representation of “how many rhombus are in a trapezoid?” for all to see (as shown here), or use the applet at
https://ggbm.at/VmEqZvke
for illustration.
[IMG]
To highlight number of groups and size of one group in the problem, discuss questions such as:
“This is a ‘how many groups of this in that?’ question. What makes 1 group, in this case?” (One rhombus.)
“How do we know whether to compare the remainder to the rhombus or the trapezoid?” (Since a rhombus makes 1 group, we need to compare the remainder to the rhombus.)
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.
Speaking, Listening: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. After a student shares their reasoning for whether they agree with Diego or Jada, ask students to restate what they heard using precise mathematical language. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


aligns: no