Grade 3 Multiplication & Area Unit Type of Lesson Title and Objective/Description Knowledge & SBAC Claim Prerequisite • Basic concept of multiples of 2’s and 5’s Knowledge: • Finding the sum of objects in an array (up to 5x5) using repeated addition Standards: Operations and Algebraic Thinking 3.OA.1, 3, 4, 5, 7 Represent and solve problems involving multiplication and division. 1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7. 3. Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. 4. Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = ÷ 3, 6 × 6 = ?. Understand properties of multiplication and the relationship between multiplication and division. 5. Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one Suggested Time Frame Math Practice Embedded 6 Weeks MA T1
can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) 7. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-­‐digit numbers. Measurement and Data 3.MD.5a, 5b, 6, 7a, 7b, 7c, 7d Geometric measurement: understand concepts of area and relate area to multiplication and to addition. 5. Recognize area as an attribute of plane figures and understand concepts of area measurement. a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. 6. Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). 7. Relate area to the operations of multiplication and addition. a. Find the area of a rectangle with whole-­‐number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. b. Multiply side lengths to find areas of rectangles with whole number side lengths in the context of solving real world and mathematical problems, and represent whole-­‐number products as rectangular areas in mathematical reasoning. c. Use tiling to show in a concrete case that the area of a rectangle with whole-­‐number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning. d. Recognize area as additive. Find areas of rectilinear figures by MA T2
decomposing them into non-­‐overlapping rectangles and adding the areas of the non-­‐overlapping parts, applying this technique to solve real world problems.(ALL MAJOR) ToK: Concepts Introduction to “Groups of”: IMP Activity “Groups of Students” Students will understand the idea of “groups of things” by finding different Claims: 1, 3 ways to arrange 12 students standing at the front of the class. Standards 3.OA.1, 3.OA.5 ToK: Arrays to Groups Memorization, Students will use multiple representations of repeated addition leading to Procedure, formally writing multiplication equations. Standards 3.OA.1, 3OA.3 Concept, RK Claim:1,3 ToK: Practicing -­‐ Arrays, Groups of, Number Line, and Repeated Addition Procedure, Sentences as Multiplication Equations Standards 3.OA.1, 3OA.3 Concepts Practice Multiplication Facts through 100 starts and continues through the Claim:1 year. Standard 3OA.3 Introduce multiplication table. ToK: Deriving the Area of a Rectangle – IMP Activity: “Where Did You Get That Concept Formula?” Students will be introduced to the concept of area and one square unit using Claim:1 grid paper and one-­‐inch foam tiles. Students will derive the area of a rectangle by building various rectangles using one-­‐inch foam tiles and noticing the relationship between area and the dimensions. Standards 3.MD.5a-­‐b, 3.MD.6, 3.MD.7a-­‐c ToK: IMP Activity: Area Model Multiplication Concepts, Students will see area multiplication as another representation of 1 hour 1,3 3-­‐4 hours 1,7,8 1-­‐2 hour 1,7,8 2 hours 1,2.7 3 hours 1,2,7 MA T3
Procedures, Relational Knowledge Claims: 1,2,3 ToK: Relational Knowledge Claim: 1,2,3 ToK: Memorization, Procedure, Concepts Claim:1 ToK: Relational Knowledge Claim: 1,2,3 ToK: Concepts, Procedures Claims: 1,2,3 ToK: Concepts Claims: 1,2,3 ToK: Relational multiplication by making area models and writing the same multiplication problem in other representations (number line, groups of, equation etc). Students will also see that equations can be written in different order without changing the meaning. Standards 3.OA.5a-­‐b, 3.MD.6, 3.MD.7a-­‐c Problem Solving with Area & Multiplication Introduction to CGI word problems (product unknown, group size unknown (partitive or fair share division), and number of groups unknown (quotitive or measurement division); students can choose a method to solve. Standard 3.OA.4 Multiplication Practice with Multiple Representations Standards 3.OA.1, 3OA.3, 3.MD.6, 3.MD.7a-­‐c Continue multiplication table. Problem Solving with Area & Multiplication Continuation of CGI word problems (product unknown, group size unknown (partitive or fair share division), and number of groups unknown (quotitive or measurement division) using bar modeling and tape diagrams. Standard 3.OA.4 Practicing Area Model: IMP Game -­ Hole in 100 Game Standards 3.MD.6, 3.MD.7a-­‐c 1-­‐2 hour 1,2,3,7 2-­‐4 hours 1,2,7 2-­‐3 hours 1,2,3,7 1 hour 1,2,6 Distributive Property & Area Multiplications: IMP Activity “Area to Distribute with 7x9” standard 3.MD.7b-­‐c 2 hours 1,2,3,7 Problem Solving with Distributive Property standard 3.MD.7b-­‐c 1 hour 1,2,3,7 MA T4
ToK: Relational Knowledge Claim: 1,2,3 ToK: Relational Knowledge Claim: 1,2,3 ToK: Concepts, Procedures, Relational Knowledge Claims: 1,2,3 ToK: Concepts, Procedures, Relational Knowledge Claims: 1,2,3 ToK: Concepts, Procedures, Relational Knowledge Claims: 1,2,3 ToK: Concepts, Procedures Claims: 1,2,3 Problem Solving with Distributive Property standard 3.MD.7b-­‐c Continue multiplication table (and throughout the rest of the year). Problem Solving Continuation of CGI word problems (product unknown, group size unknown (partitive or fair share division), and number of groups unknown (quotitive or measurement division) using bar modeling and tape diagrams. Standard 3.OA.4 Finding the Area of a Rectilinear Figure: IMP Lesson – “ Slice & Dice “ Students will learn properties of the area of rectilinear figures through a series of explorations where they put together and take apart rectangles to see that the total area will remain the same. The number of square units does not change just because the rectangular figures are connected or disconnected. Standard 3.MD.7d Practice Finding Area of Rectilinear Figures Standard 3.MD.7d 1 hour 1,2,3,7 2-­‐3 hours 1,2,3,7 2-­‐3 hours 1,2,3,7 1 hour 1,2,3,7 Problem Solving with Area of Rectilinear Figures Standard 3.MD.7d 1 hour 1,2,3,7 Factor Game -­ Four in a Row Students will make connections with multiplication through a game called Four In a Row. Standards 3.MD.6, 3.MD.7a-­‐c Throughout the rest of the school year 1,2,6 MA T5
Codes: RK-­‐ Relational Knowledge Claim 1: Concepts/Procedures C-­‐Conceptual Knowledge Claim 2: Problem Solving P-­‐Procedural Fluency Claim 3: Communicating & Reasoning M-­‐Memorization Claim 4: Model and Data Analysis Unit Plan Needs: Practice: • Practicing -­‐ Arrays, Groups of, Number Line, and Repeated Addition Sentences as Multiplication Equations • Multiplication Practice with Multiple Representations • Practice & Problem Solving Finding Area of Rectilinear Figures • Multiplication Facts – Fluency within 100 Problem Solving • Problem Solving with Area & Multiplication (CGI word problems) • Problem Solving with Distributive Property • Problem Solving -­‐ Finding Area of Rectilinear Figures MA T6
Teacher Directions This is meant as a quick exercise to introduce students to the idea of “groups of” before starting multiplication. Task 1: • Use a think-­‐pair-­‐share for the brainstorm question on the student page. • Choose 12 students to come up to the front of class and stand in a long straight line. Task 2: • Guide students to make 12 quick happy faces to represent each of the 12 students at the front of the class. o Answer for student page: “Here we have one group of twelve students.” • Guide students to draw a line between each happy face separating them into 12 groups of 1. Once lines are drawn, ask students if they can think of another way to look at the group using the numbers 12 and 1. o Answer for student page: “We could also think of the line as twelve groups of one student.” Task 3: • Now separate the students into 4 groups of 3 students (instead of being in one long line side by side). Ask students to represent this new arrangement with circles and happy faces, then fill in the blanks with “4 groups of 3 students is the same as twelve students.” See sample below. • Then challenge students to come up with another arrange the students using the same numbers 3,4, and 12. This would then be “4 groups of 3 students”. Task 4: Use a think-­‐pair-­‐share to brainstorm 2 groups of 6 and 6 groups of 2. !
IMP Activity: Groups of Students
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MA T7
Teacher Directions Multiple Representations (problems 1-­‐4): • Guide students through number one asking questions and allowing students to think-­‐pair-­‐share. Sample questions include, “How could we write an addition sentence that uses the same number added together over & over when looking at the picture of the eggs?” “Who can think of a way to show our repeated addition sentence on our number line? What numbers should I use on the number line? Why?” “Does anyone see some groups of eggs?” “Can you show me the groups by circling them?” etc. Have students discuss and show their ideas under the document camera. Put both ways to represent the array in the boxes. Ask guiding questions to help the students see both ways (ex: What would happen if I turned my paper this way?” etc) • Sample responses below Array of Eggs: Repeated Addition Sentence(s) 2+2+2=6
3+3=6
Number Line 0 1
2
3
4
5
6
Words: I have ____3___ groups of ____2____eggs. OR I have ____2___ groups of _____3___eggs. Completing the Table: • Students will copy their addition sentences into the table form questions 1-­‐4. Tell students to NOT yet put anything in the multiplication column. • Ask students to then look at the dot array. The big idea here is that writing the addition sentences can be tedious when there are many objects to count. • After the group has agreed that writing the addition sentences for the dot arrays is a hassle, guide students through the steps to write their multiplication sentences/equations while introducing the terms and symbols. • Students will then have the opportunity to practice with a partner. The groups of triangles (3 groups of 2) and dots (4 groups of 4) are more clear cut as far as “groups of”, but the paperclips and arrows may be seen in different ways by students. Encourage different methods and have students explain & share. • Paperclips: could be “6 groups of 3” or some students may see it as “ 2 groups of 9” or “3 groups of 6” since they arranged like an array. • Arrows: “5 groups of 3” or “3 groups of 5” !
IMP Activity: From Arrays to Groups
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MA T8
Teacher Directions: Multiplication Chart *In order to help students skip count, repeated practice is necessary and can be done through songs, skip counting competitions between groups, videos (here is a sample of counting by 8’s: https://www.youtube.com/watch?v=d2fkzKP2A3M); you can also challenge students to come up with their own songs and videos. *This is only suggested and can be modified as needed. The days suggested do not have to be an entire math class period. It could be used as the first or last 20 min of math within a day. Days 1-­‐2: Pass out a completed multiplication chart and ask students what they notice. There are many things the students may notice (sample responses: identity property (all number multiplied by one is itself); Commutative property of multiplication 3x4=12 and 4x3=12 are both on the table but in different locations); each row are multiples of the factors listed vertically and horizontally (skip counting); odd and even patterns (even x even = even), squares are along the diagonal in the middle of the table and the other numbers match up if we were to fold the table along that diagonal. Days 2-­‐3: Learning to create a multiplication chart: Note: One objective for this unit includes teaching the students to draw a 10 x 10 multiplication chart to use in all problems as well as to draw and then use on exams. The intent is to train the students to draw these quickly, so they can be done in under 5 minutes. This goal is in lieu of trying to get the students to memorize the multiplication facts for those who struggle. Pass out the multiplication chart that is half way completed. Have the students complete the chart independently. Some methods students can use are as follows: 1) add on across a row; i.e., 1, 2, 3, or 2, 4, 6; 2) fill in facts they have memorized, such as the 1’s, 2s 5’s and 10’s and then use counting to fill in the holes. Whatever method you suggest, ask students why they think the grey boxes are filled in and how that might help them with the white boxes. Help them see that they only need to figure out the answers for ½ of the products and then they can copy on the left diagonal half. Practice having students complete the multiplication chart that ishalf way filled out several times. Day 4-­‐5: Pass out multiplication chart with factors only. This time, the table does not have any products filled in. Encourage the students to use any patterns they found in previous days (especially the idea of only solving the top half of the diagonal) as well as counting on, etc. to complete the table. Practice having students complete the multiplication chart with the factors only. After they have had some experience, students can then record their time, the goal being to continuously better their time. Day 6-­‐7: Pass out blank multiplication chart. This time, the table does not have anything filled in. (Note that we are moving towards the students being able to create one on their own.) Encourage the students to use any patterns they found in previous days (especially the idea of only solving the top half of the diagonal) as well as counting on, etc. to complete the table. Have students try to beat their previous time. Irvine Math project Activity: Multiplication Chart MA T9
Multiplication Table – Creating One On Your Own
Directions: Draw your own multiplication chart. To do this, you will need a table that has 11 columns and 11 rows. Leave the top corner box blank, and then begin to write the numbers 1-­‐10 in each of the top row boxes. Write the numbers 1-­‐10 in each of the boxes going down the first column as well. Then complete the multiplication chart below from memory or by counting or using another pattern you have discovered. (Remember that you only need to solve for answers on the top diagonal and then you can copy those to the left bottom diagonal). Irvine Math project Activity: Multiplication Chart MA T10
Teacher Directions - *Save each students’ cm grid paper for later use with “Slice and Dice”
Materials:
1-inch square tiles (approx. 50 per student)
cm grid paper – see attached (one per student)
Tasks 1 & 2:
Pass out 1-inch square tiles and the packet of “grids” for each student. Have participants start on task 1,
making sure that students use the tiles and that everyone in the groups has sketched a rectangle with
different dimensions. Explain and model the task. Ask students to use the blank grid sheet to draw a
rectangle. Each person in the group should make a different rectangle. Cover the rectangle with square
inches (tiles) and decide how many total square tiles cover your rectangle. Record this in the “area”
column. Then identify what is the length and width and record this in the table. Share each other’s
information and include it in the table below.
This is the time to introduce the word “area” when explaining the total number of square tiles covering the
rectangles. Put the word “area” on the word wall. When you see that the groups are close to finishing their
first task, ask for volunteers to give their dimensions. Make a chart in the front of the room with their
information, and lead the discussion.
Have students use think-pair-share to discuss their responses to questions 1 and 2 and then call on
students to share ideas.
Important Ideas for Class Discussion
• Why does a 3 x 2 make sense for the number of tiles, i.e. area? (because you have 3 groups/rows of 2,
hence this is multiplication )
• So why does it make sense to have l x w as a representation of area? (because you have l groups of w;
hence this is the definition for multiplication)
• Length and width are called “inches” because we are only talking about a linear dimension (we could
use a ruler to measure l and w); area is inches squared because we are talking about how many one
inch squares fit inside the figure.
• Point out the exponent in the Area column (square in) and discuss the idea of 2 dimensions for area vs
1 dimension for l and w.
IMPORTANT: Point out that now they have a formula for finding the area of a RECTANGLE using
the definition of multiplication. Have the students record their formula in the text box on their page.
Task 3: Practice *Note: Save these rectangles for activity for area of rectilinear figures.
Pass out the cm graph paper to each student and have each student draw 4 rectangular figures (squares are
rectangles so those can be included too). Explain that our units are now different. Ask some questions
such as, “Does it look like cm are smaller or bigger than in?” Why do we call the Area centimeters
squared?” “Why do we call l and w centimeters?” etc. Have students complete the table on the student
page using their rectangle information. (ans: l and w are each one dimension whereas Area includes 2
dimensions (l AND w) or the number of unit squares that cover the figure).
IMP Activity: Where Did That Formula Come From?
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MA T11
Guide students through the questions using think-pair-share. Answers: Square #5 has a length of 1cm and
a width of 1cm and an area of 1 square cm.
Sketch #5 here:
Why is the area of Square #5 just 1 square cm? Explain Only one square is needed to cover the area or
space between l and w.
What if rectangle #5 had a length of one foot (ft) and a width of one foot (ft)? We would then say Square
#5 has a length of 1ft and a width of 1ft and an area of 1square ft .
What if rectangle #5 had a length of one meter (m) and a width of one meter (m)? We would then say
Square #5 has a length of 1m and a width of 1m and an area of 1square m.
What if rectangle #5 had a length of one unit (un) and a width of one unit (un)? We would then say
Square #5 has a length of 1un and a width of 1un and an area of 1square un.
What is the same about 1 square cm vs 1 square foot? Side length of one; area of one; each one square
unit; both squares etc
What is different about 1 square cm vs 1 square foot? Different in size; different units of measure; If using
the same unit of measure, the side lengths would be different etc
IMP Activity: Where Did That Formula Come From?
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MA T12
Teacher Directions: Materials: • Student Page for Area Model Multiplication • Optional: Graph Paper Objective: Students will build off of their experience deriving the formula for the area of a rectangle to represent multiplication problems as area with a grid. Students will also use additional representations to model the multiplication problem such as arrays, number line, with words, or equations. Directions: • Ask students to recall the previous activity using squares to build a rectangle and discover that the area of a rectangle is length times width. • Pass out the student activity page and ask students to think-­‐pair-­‐share to determine how we could draw a picture using area model to represent 3x5 in the space provided for #1. • Walk around to observe. then have a student share by bringing their paper up to the document camera. Ask the student to explain how they knew to draw a 3x5 grid. Sample questions: “Where do you see the three? Where do you see the five? How many total boxes do you see in the grid? What is the total number of boxes called? (ans: Area) Is there a faster way to determine the total other than counting each box individually? Explain.” Etc. • *Note: It will take some practice for students to draw the area models. Students can also be given graph paper to scaffold the process of learning to make the area model grids. • Ask students if someone had another picture that was a little different. Ask students if the orientation of the grid matters (ans: No. Either direction is correct – see example below.) 3 3 5 IMP Activity: Area Model Multiplication
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MA T13
•
Students will now write equations two ways to represent the area and practice commutative property of multiplication. • Ask students to think-­‐pair-­‐share how they might fill out the equations. Ask leading questions such as, “What could we call the length of this rectangle, what could we call the width of the this rectangle, what is the area? • *Teacher Note: The length does not have to be the longest side of the rectangle. It can be either. Example for #1: 3 x 5 = 15 length x width = area 15 = 5 x 3 area = width x length • Within the same multiplication problem the length and width should stay consistent though. In the above example, the length is the shorter side (l = 3), but the length is consistent in equations #1 and #2 (again, both l = 3). • Allow students to work independently or with a partner to complete the rest of the problems on pages 1-­‐2. • Then ask students to choose four of the following options to represent the remaining problems on pages 3-­‐4 (see example of #7 below) bringing together their previous work with arrays, groups of, and number line with the new additions of area model and equations using A = l × w . Encourage students to use a variety of representations, not the same four over and over. • Students will then summarize with the sentence frame by writing their favorite method and why. Array Groups Of I have 11 groups of 3. 11 × 3
Number Line Area Model 3
0
3 6
9
3!
3!
!
!
12 15 18 21 24 27 30 33
!
3!
!
3!
!
3!
!
3!
!
3!
!
3!
!
3!
IMP Activity: Area Model Multiplication
11
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Teacher Directions:
Materials:
• Student page (one per person)
• Dice (pair of dice per pair of students)
How to Play:
• Each students is filling in their OWN grid individually.
• Take turns with a partner rolling a single die.
• Each time you roll you must fill in one of the factors of one of the multiplication sentences.
It can be ANY of the spaces provided for factors. You do not have to fill in the factors one
after the other. Be strategic to try and fit as many rectangles in the grid as possible.
• When both factors of a multiplication sentence have been filled in, write the product and
draw a rectangle to represent the sentence. Write the sentence in the drawing.
• If a student writes a multiplication sentence with a rectangle that will no longer fit, then the
student cannot use that rectangle.
Example: A possible board after 6 rolls.
x
=
.
2 x 4
=
8 .
10
2x4=8
3 x 6
= 18
.
5 x
=
.
1 x
=
.
Filled Area =
Missing Area =
10
3 x 6 = 18
.
.
How to Win:
• The goal is to fill as much of the whole area (100) as possible.
• The winner is the partner who has the largest total area (closest to 100 square units).
IMP Activity: Holes in 100
MA T15
Teacher Directions Materials: • Area to Distribute 7x9 student page (one per student) • Colored pencils or markers (one set for teacher & each group/pair) Task: • Ask students to think-­‐write-­‐pair-­‐share to answer the question, “How could I multiply 7 x 9 without using the digit 9?” • *Note: resist the temptation to give the students hints or an example prior to giving the 7x9 question. • Enforce 2-­‐3 minutes of silent think time. During the silent time, ask students to put up 1 finger to their chest when they have one idea, 2 fingers for 2 ideas and so on. • After the silent time, have students write down their methods on some scratch paper (no talking yet). • Pair: Allow students to share their methods with a partner or group. If sharing within groups, be sure to use roundtable method where each person shares their thinking going around in a circle starting with the person with the longest hair or the oldest etc. Ask pairs/groups to come up with 2-­‐3 methods they want to share out with the class, designating their top choice, but having backup if another group says their idea. • Share: Randomly call on students using index cards or popsicle sticks to share ideas. Scribe the students’ ideas on poster paper or on the white board (correct & incorrect responses). Write name or group number next to each idea. • Possible correct ideas: o 7+7+7+7+7+7+7+7+7 o 7(6) + 7(3) o 7(7+2) o 7 x 3 x 3 • Possible incorrect ideas: o (7x10)-­‐1 o (7x8) + 1 o 7x9 • *Note: Students have not formally been introduced to order of operations. When they share their ideas, be sure they clarify what they mean. A student could have a correct idea that needs clarification. For example (7x8) +1 is incorrect, but if the student says they broke apart 9 into 8 and 1 and they are seeing it as 7x8 + 7x1, that is correct. The emphasis on order of operations is learning the order in a context such as this task. • Pass out student page with 7x9 grids. Tell students we will be verifying the various ideas using the grid paper, but first we must all agree on the area of the grids (A = 63 square units) . Ask students to determine the area of the Irvine Math Project Activity: Area to Distribute 7x9
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MA T16
•
•
•
•
•
•
7x9 grids on their paper. Have students share their answers and explain how they know. Now tell students that as a class we will go back and verify if the scribed responses are correct or incorrect using the area model. Revisit each response given earlier on the poster paper and “prove” or verify it is correct or not. For example, show 7(6) + 7(3) as 7 groups of 6 using one color to outline or shade, then 7 groups of 3 using another color to outline or shade. If students give all correct responses, add a couple incorrect ones to the list to investigate. You will need to add squares to the grid for some responses (ex: (7 x10) -­‐7; here you would add a row at the top or bottom of the grid so that you can shade 7 groups of 10) then you will show subtracting 7 boxes by circling it, crossing it out etc. As you verify a few, ask clarifying questions. Ex: “Does ___’s method have a total area of 63? How do you know?” “What do you think ___was thinking when he came up with (7x10)-­‐1? How could we adjust his method to make it correct?” If there are more methods remaining on the list, have students work independently or in pairs on their grids to represent the remaining methods. *Note: 7x3x3 can be thought of as 7 groups of 3, 3 times. It can also be thought of as 7 groups of 3x3. You can shade 6 3x3 grids, but due to the limitations of the 7x9 grid, the 7th 3x3 grid does not fit (only an additional 3 one by three grids). Use this as an opportunity to discuss with students the different meanings & representations of 7x3x3. Finish the lesson with the practice & problem solving tasks on the last page. Sample answers for the skateboard ramp problem include: o 6 8x1 pieces of rubber o 3 8x2 pieces of rubber o 2 8x3 pieces of rubber o 1 8x2 plus 1 8x4 o 1 8x1 plus 1 8x2 plus 1 8x3 and so on. Irvine Math Project Activity: Area to Distribute 7x9
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MA T17
Teacher Directions Materials: • Scotch tape (one per pair) • Scissors (one per pair) • “Where Did That Formula Come From?” student pages distributed to each person Tasks 1 & 2: • Students will use their “Where Did That Formula Come From?” student pages and copy the centimeter data from page 2 of “Where Did That Formula Come From?” into the table at the top of page one of Slice & Dice. • Guide students to find the sum of all 4 areas and to write an addition sentence. • Model how to tape the 4 rectangles together as seen in the picture on the student pages in a creative way. Key points are that there should be NO overlaps , NO gaps, and small squares should LINE UP with other small squares. Model how to have a partner help line up the rectangles while the other person tapes. Tape the figures together well. • Then use a think-­‐pair-­‐share for students to predict if the area of their new figure is the same as the sum of the areas of the 4 separate rectangles. Try to have a yes and no answer share and justify. • Then have student count up the squares in the new figure and have their partner verify their final answer. Then have students summarize the big idea (area stays the same when you are joining together figures without gaps or overlaps; original rectangles did not change size/area just because they were taped together) in the remaining questions. Task 3: • Now students will cut up the figures again into NEW RECTANGLES, but it has to be DIFFERENT rectangles/squares than they had originally. 1) Original 2) New Figure Taped Together 3) New Cuts IMP Activity: Slice and Dice
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MA T18
*For the teacher modeling the tasks and big ideas at the end, cut out new rectangles each time to arrange and rearrange to show the progression. Students’ rectangles will be taped together, then cut, then taped again, but it will be nice as the teacher to show all 4 steps side by side (e.g. 4 original rectangles, new figure, 4 new rectangles, and final figure side by side). • Have students write down the dimensions of the new rectangles in the table. Big Idea: (sample/possible responses) We can put together and break apart rectangular figures and the total area will remain the same. The number of square units has not changed just because the rectangular figures are connected or disconnected. Task 4: • Here student will practice putting it all together by breaking apart figures to find the total area of the figure. Validate any and all correct methods. Circulate as students work to find different representations. Have student share their various methods as there are many correct ways to break down the figures into different combinations of rectangles. Task 5: • Allow students to work independently first to brainstorm ways to solve, then allow partners to talk and share their thinking. Have pairs come up to the document camera to show various methods to solve. Always ask “did someone see it a different way?” to encourage multiple methods to solve for the total area. IMP Activity: Slice and Dice
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MA T19
Teacher Directions Materials Counters (need two different colors) (about 20 per student or team) Overview: Students will play a game to practice multiplication of whole numbers and fractions by choosing two factors and placing a counter on a square in an attempt to cover 4 squares in a row. Directions This is a game for two players or two teams and can easily be played with the whole class by dividing the class into two groups and placing the game board on the document camera. The object of the game is to be the first team to get 4 in a row. The first team to go chooses two factors from the bottom of the board and calls out the product. Place a counter on each of the factors and place a counter on the square representing the product – each team is a color on the board (note that the group or person choosing the factor must explain which square to cover, demonstrating their understanding of multiplication or division). The next team now must choose one of the markers to move to another factor and call out the product. They may only move one of the markers in the advanced version of the game, but to scaffold the game, start by allowing students to move both. Note: Both counters may be placed on the same factor (e.g., 2 x 2 = 4). Teams should quickly realize that they must not only think about how to get 4 in a row for themselves, but also be careful to block the other team and avoid leaving the paperclips on factors that the other team needs to complete their 4 in a row. IMP Activity The Factor Game- Multiplication & Division
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MA T20
MA T21
MA T22
MA T23
MA T24