standard_description: Interpret multiplication as scaling (resizing), by:
- Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
- Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
problem_activity: Narrative
The purpose of this activity is for students to use a bar graph to compare two quantities and describe the methods they use to find the unknown difference. Monitor for students who draw on the graph and describe ways of finding the difference by counting on or counting back. If students draw on their graph or do not discuss both counting methods during the activity, create and display work so that each method can be discussed during the synthesis. The discrete segments of the bar graph are used to elicit these counting methods (MP5), however, some students may use addition or subtraction, including known sums, to find the difference. Encourage these students to connect their methods to the counting methods shared in the synthesis.
MLR8 Discussion Supports.
Provide all students with an opportunity to speak and to practice using comparison statements. Invite students to chorally repeat statements that use “more” and “fewer” in unison 1–2 times.
Advances: Speaking, Listening
Launch
Groups of 2
Display the image (graph with no scale).
“A group of third grade students were asked, ‘What pets do you have?’ Their responses are shown in the bar graph."
"What do you notice about the data in the graph? What do you wonder?”
1 minute: quiet think time
1 minute: partner discussion
Monitor for students to say: “More students have cats than have rabbits,” or wonder, “How many more students have cats than have rabbits?”
Share and record student responses.
If students do not make statements using “more” or “fewer,” display:
“There are more _____ than _____.”
“There are fewer _____ than _____.”
“How could you complete the sentences to make them true statements about the graph?”
1 minute: quiet think time
1 minute: partner discussion
Share responses.
Activity
“You noticed that more students have cats than have rabbits. Your job now is to figure out how many more students have cats than have rabbits. Think about two different ways you can find the answer and record them.”
3–4 minutes: independent work time
2–3 minutes: partner discussion
Monitor for a student who uses a counting on method and a student who uses a counting back method.
Student Facing
A group of third grade students were asked, "What pets do you have?" Their responses are shown in the bar graph.
What do you notice? What do you wonder?
[IMG]
Their responses are also shown in this bar graph.
[IMG]
How many more students have cats than have rabbits? Show two ways to find the difference.
Student Response
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Activity Synthesis
Invite previously identified students to share how they found the difference. If methods are only explained verbally, consider asking, “Can you show us on the bar graph how you could count on (or count back) to find how many more?”
Record student methods with an equation. For example, for students who describe counting on, write “8 + 9 = 17.”
"How are their methods similar? How are their methods different?" (One method starts with the smaller number of pets and counts on to the larger number, like adding 9 more. The other starts with the larger number of pets and counts back to the smaller number, like subtracting 9.)


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