Elementary and Middle School Mathematics

GLOBAL

EDITION

Elementary and Middle School

Mathematics

Teaching Developmentally

TENTH EDITION

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Reasoning Strategies for Addition Facts

225

FIGURE 9.4 Situations for doubles facts.

6 + 6 8 + 8 4 + 4 5 + 5 7 + 7 9 + 9

Doubles

There are ten doubles facts from 0 + 0 to 9 + 9. These facts are foundational, provided the basis for near doubles strategy. Many students find doubles easier to grasp than other facts. + 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 2 4 6 8 10 12 14 16 18

However, all students, and especially students with disabil-ities, can benefit from using and creating picture cards for each of the doubles as shown in Figure 9.4. Story problems can focus on pairs of like addends: “Alex and Zack each found 7 seashells at the beach. How many did they find together?”

CCSS-M: 1.OA.C.5; 1.OA.C.6; 2.OA.B.2

Double Trouble

Read Double the Ducks (Murphy, 2002), a story that begins with 5 ducks each bringing home a friend. Have students work with a partner. Give each pair a set of counters. The first partner selects some of those counters to be the ducks. Their partner tells how many ducks there will be when each duck brings a friend. To support student reasoning, ducks (counters) can be placed on a ten-frame.

Activity 9.4

CCSS-M: 1.OA.C.6; 2.OA.B.2

Calculator Doubles

Students work in pairs with a calculator. Students enter the “double maker” 12*2 into the calculator. One student says a double—for example, “Seven plus seven.” The other student presses 7, says what the double is, and then press = to see the correct double (14) on the display. The students then switch roles and reset the calculator 12*2. For ELs who are just learning English, invite them to say the double in their native language or in both their native language and English. (Note that the calculator is also a good way to practice + 1 and + 2 facts.)

ENGLISH LEARNERS

Perhaps the most important strategy for students to know is the combinations that equal 10. It is a foundational fact from which students can derive many facts (Kling, 2011). Consider story situations such as the following and ask students to tell possible answers.

There are ten boys and girls on the bus. How many girls and how many boys might be on the bus?

There are many children’s books focused on the concept of 10. While they might be counting books, you can incorporate questions like, “How many more to equal 10?” Such books could be used as a context for Activity 9.6.

FIGURE 9.5 Combina-tions of 10 on a ten-frame.

6 and 4 is 10.

CCSS-M: 1.OA.B.4; 1.OA.C.6; 2.OA.B.2

How Many More to Equal 10?

Place counters on one Ten-Frame (see Figure 9.5) and ask, “How many more to equal 10?” This activity can be repeated using different start numbers. Eventually, display a blank ten-frame and say a number less than 10. Students start with that number and complete the “10 fact.” If you say, “four,” they say, “four plus six equals ten.” This can be completed as whole class or with students working with a partner. Students who are still in phase 1 of learning the facts (using counting strategies) or students with disabilities may need additional experience or one-on-one time working on this process.

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Blackline Master: Ten-Frame

Activity 9.6

10

+

Adding onto 10 is not an official basic fact, but in preparation for the Making 10 strategy, stu-dents must know Combinations of 10 and 10 + facts. Using a full ten-frame and a par-tial ten-frame can help students with this group of facts. Just like with one more and two more, the goal is to move from counting on to “just knowing.” A number line or hundreds chart can help students notice the relationship in these sums that will help them move beyond counting.

10 10 10 10 10 10 10 10 10 + 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

Making 10

Reasoning Strategies for Addition Facts

227

This reasoning strategy is extremely important and is heavily emphasized in high- performing countries (Korea, China, Taiwan, and Japan) where students learn facts sooner and more accurately than U.S. students (Henry & Brown, 2008). Yet this strategy is not emphasized enough in the United States. A study of California first graders found that this strategy con-tributed more to developing f luency than using doubles (even though using doubles had been emphasized by teachers and textbooks in the study) (Henry & Brown, 2008).

The Making 10 strategy can also be applied to larger numbers. For example, for 28 + 7, students can make 30, seeing that 28 + 7 = 30 + 5. Thus, this reasoning strategy deserves significant attention in teaching addition (and subtraction) facts.

Quick images or manipulating Double Ten-Frames (see Blackline Master 15) can help students develop the Making 10 strategy. For example, cover two ten-frames with a problem, like 6 + 8. Ask students to visualize moving counters from one frame to fill the other ten-frames and explain their thinking. Move to doing this mentally. Activities 9.7 and 9.8 are designed for this purpose.

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Blackline Master: Double Ten-Frames

CCSS-M: 1.OA.B.3; 1.OA.C.6; 2.OA.B.2

Move It, Move It

Distribute the Move it, Move it Activity Page and a mat with Double Ten-Frame (Blackline Master 15). Flash cards with facts are placed next to the double ten-frame, or a fact can be given orally. The students cover each frame with counters to represent the problem (9 + 6 would mean covering nine places on one frame and six on the other). Ask students to “move it”—to decide a way to move the counters so that they can find the total without counting. Ask students to explain what they did and connect to the new equation. For example, 9 + 6 may have become 10 + 5 by moving one counter to the first ten-frame. Emphasize strategies that are working for that student (5 as an anchor and/or Combinations of 10 and/or Making 10).

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Activity Page: Move It, Move It

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Blackline Master: Double Ten-Frame

Activity 9.7

CCSS-M: 1.OA.B.3; 1.OA.C.6; 2.OA.B.2

Frames and Facts

Make Little Ten-Frame Cards and display them to the class on a projector. Show an 8 (or 9) card. Place other cards beneath it one at a time as students respond with the total. Have students say aloud what they are doing. For 8 + 4, they might say, “Take 2 from the 4 and put it with 8 to make 10. Then 10 and 2 left over is 12.” Move to harder cards, like 7+ 6. After modeling with whole group, place students with partners, trading off who is

Activity 9.8

STUDENTS with SPECIAL

FIGURE 9.6 Frames and Facts Activity.

Frames and Facts

5

+ 7

6

+ 5

4

+ 6

7

+ 4

Use 10

Use 10 is recently researched reasoning strategy (Baroody, Purpura, Eiland, & Reid, 2014; Baroody et al., 2016). The thinking process of Use 10 is different than Making 10, and it works for the same facts as highlighted above. You start with 10 + fact and adjust the answer. For example, for 9 + 6, a student thinks, 10 + 6 is 16, 9 is one less than 10, so the sum is also one less, 15. Notice this does not require decomposing or recomposing a number. Students who learned this strategy and practiced it while playing games, did substantially better than their peers on addition facts involving adding 8s or 9s (Baroody et al., 2016).

Using 5 as an Anchor

Using 5 as an anchor means looking for fives in the numbers in the problem. For example, in 7 + 6, a student may see that 7 is 5 + 2 and that 6 is 5 + 1. The student would add 5 + 5 and then the extra 2 from the 7 and the extra 1 from the 6, adding up to 13. The ten-frames can help students see numbers as 5 and some more. And because the ten-frame is a visual model, it may be a strategy that visual learners and students with disabilities find particularly valuable. Like Making 10 and Use 10, this strategy works for all the facts with sums between 11 and 18 (see graphic in Making 10 section).

Near-Doubles

Near-doubles are also called “doubles-plus-one” or “doubles-minus-one” facts and include all combinations where one addend is one more or less than the other. The strategy is to double the smaller number and add 1 or to double the larger and then subtract 1. Therefore, students must know their doubles before they can work on this strategy.

placing the frame and who is figuring out the sum. Ask students to record each equation (see Figure 9.6). Especially for stu-dents with disabilities, highlight how they should explicitly think about filling in the little ten-frame starting with the higher number. Show and talk about how it is more challenging to start with the lower number as a counterexample.

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Activity Page: Little Ten-Frame Cards

1 3 5 7 9 11 13 15 17 1 3 5 7 9 11 13 15 17 + 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

To develop this strategy, begin with a double fact quick image, such as 6 + 6, and follow it with a quick image of 6 + 7. Transition to using expressions, writing a doubles fact and a near-doubles, as the ones illustrated here:

4 + 4 5+ 5 6 + 6 8+ 8

4 + 5 5+ 6 6 + 5 7+ 8

Reasoning Strategies for Subtraction Facts

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Near-doubles can be more difficult for students to recognize and therefore may not be a strategy that all students find useful. In that case, do not force it.

FIGURE 9.7 Near-doubles fact activity.

Put the near-double on the double fact that helps. 6 + 6 3 + 3 4 + 4 7 + 8 8 + 8 1 + 1 9 + 9 2 + 2 7 + 7 5 + 5 5 + 4 3 + 4 7 + 6 9 + 8 5 + 6

CCSS-M: 1.OA.B.3;

1.OA.C.6; 2.OA.B.2

On the Double!

Create a display (on the board or on paper) that illustrates the doubles (see Figure 9.7). Prepare cards with near-doubles (e.g., 4+ 5). Ask students to find the doubles fact that could help them solve the fact they have on the card and place it on that spot. Use Little

Ten-Frame Cards with doubles and near doubles to help students, particularly those with disabilities, visualize the relationships. Ask students if there are other doubles that could help.

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Activity Page: Little

Ten-Frame Cards

Activity 9.9

Reasoning Strategies for Subtraction Facts

Subtraction facts prove to be more difficult than addition, perhaps because it receives less atten-tion and because it requires knowing the related addiatten-tion facts. Figure 9.1 at the beginning of the chapter lists the ways students might subtract, moving from counting to mastery. Without opportunities to learn and practice reasoning strategies, students continue to rely on counting strategies for subtraction, a slow and often inaccurate approach. Therefore, spend sufficient time working on the reasoning strategies for subtraction to help students move to phase 2 and eventually on to mastery/automaticity (phase 3).

Think-Addition

As the label implies, in this strategy students use known addition facts to produce the unknown quantity or part of the subtraction. For 13 - 8, a student thinks, “What plus 8 equals 13? Stu-dents taught this strategy do significantly better than their peers on subtraction (Baroody et al., 2016). The value of think-addition cannot be overstated; however, it requires that addition facts be mastered. If this important relationship between parts and the whole—between addition and subtraction—can be made, subtraction facts will be much easier for students to learn. One way

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Video Example 9.6